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\begin{document}
\title{Pump-probe Spectroscopy}
\author{ Nguyen A. Tuan\\
     \\
         Department of Physics\\
         University of Cincinnati \\
         Cincinnati, Ohio 45221\\[1em]}

\date{}
\maketitle
\begin{abstract}
Ultrafast spectroscopy of semiconductors is currently one of the
most exciting areas of research in condensed-matter physics.
Remarkable progress in the generation of tunable femtosecond
pulses has allowed direct investigation of the most fundamental
dynamical processes in semiconductors. This paper presents a brief
overview of the pump-probe technique, the most common form of
ultrafast spectroscopy, and several its variations which are
widely used in practice.
\end{abstract}

\pagebreak

\def\baselinestretch{1.1}

INTRODUCTION

\vspace{0.1in}

Pump-probe spectroscopy is a powerful technique for investigating
electronic and vibrational properties of variety of systems, and
has brought out extensive information into the properties of
atoms, molecules and solids. Its strengths have been exploited for
many years and recently have been strengthen when combined with
the picosecond and femtosecond laser pulses to provide new
insights into completely different aspects of semiconductors.

%It is necessary here to emphasize the role of ultrashort laser
%pulses. One important application domain of ultrafast lasers is
%the behavior analysis of a sample perturbed by such a light pulse.
%Generally, during the perturbation, it is difficult to make
%measurements. It is usually necessary to wait till the end of the
%perturbation before analyzing the consequences. However, if some
%processes are very fast compared to the duration of the
%perturbation, these processes will be hidden so that only those
%processes which are longer than the pulse duration could be
%observed. Unfortunately, those processes can be the consequences
%of the faster ones. Therefore, the shorter the perturbation, the
%easier to understand the origin of the faster processes.

\vspace{0.1in}

GENERAL PRINCIPLES

\vspace{0.1in}

As the most common form of ultrafast optical spectroscopy, the
pump-probe method needs at least two ultrashort light pulses. The
first one, the so-called "pump" pulse, perturbs the sample at time
$t=0$. The second one, the so-called "probe" pulse, delayed with
respect to the "pump" pulse, crosses the perturbed sample and acts
as a probe \cite{ref1, ref2}. In the simplest form of pump-probe
spectroscopy (so-called \it degenerate \em pump-probe
spectroscopy), the output pulse train from an ultrafast laser is
divided into two. The semiconductor sample under investigation is
excited by the pump pulse and the changes induced in the sample
are probed by the probe pulse which is suitably delayed with
respect to the pump by introducing an optical delay in its path.
Some property related to the probe (e.g. reflectivity, absorption,
Raman scattering, luminescence) is then monitored to investigate
the changes in the sample produced by the pump. In \it
non-degenerate \em pump-probe spectroscopy, one uses two
synchronized lasers at different wavelengths, or a laser and a
synchronized white light continuum. This increases the versatility
of the technique enormously by allowing a determination of the
changes induced at photon energy different from that of the pump.
Figure 1 represents a typical schematic diagram of a pump-probe
experiment.
\begin{figure}[htb]
\centerline{\psfig{figure=Fig11.eps}}

\caption{\em{Schematic for a generic setup for a pump-probe
experiment. This setup can be used for pump-probe transmission or
reflection spectroscopy, FWM spectroscopy, or luminescence
correlation spectroscopy as shown \cite{ref1}.}}
\end{figure}
\vspace{0.1in}

VARIATIONS OF PUMP-PROBE SPECTROSCOPY

\vspace{0.1in}

Many different pump-probe spectroscopy techniques have been
developed based on different techniques of linear optical
spectroscopy. The following sections will briefly discuss them.

\vspace{0.1in}

\bf 1. Transmission/Reflection Spectroscopy. \rm

Perhaps the simplest of pump-probe methods is measurement of
transmitted or reflected probe. The probe is typically much weaker
than the pump, and the spot diameter of the probe at the sample is
considerably smaller than that of the pump so that it probes a
region of uniform photoexcited density. For the transmission
spectroscopy (similarly to reflection spectroscopy), one generally
measures the \it change \em in the transmitted probe pulse energy
induced by the pump as a function of the time delay between the
two pulses by chopping the pump beam and using lock-in amplifier.
Some experiments may require that the pump and probe beams are
chopped at different frequencies, and the change in transmitted
probe is detected at, for instance, the sum of the two chopping
frequencies.

The data are typically presented in the form of normalized
Differential Transmission (DT) given by $\Delta
T/T_{0}=(T-T_{0})/T_{0}$, the change in transmission $\Delta
T=T-T_{0}$ induced by the pump pulse divided by the transmission
of the probe in the absent of the pump, $T_{0}$. The Differential
Transmission Spectrum (DTS) can be obtained either by varying the
wavelength of the probe laser or by spectrally analyzing the
transmitted probe energy with a spectrometer if a broadband
ultrafast source (e.g. white light continuum) is used as the
probe. The use of a multichannel as diode array or CCD camera is
generally preferable in the latter case. Although one usually
measures the total transmitted probe energy, the transmitted pulse
shape can be measured by a suitable fast detector or a nonlinear
gating technique \cite{ref1}.

%The strength of this pump-probe spectroscopy technique is its
%ability to obtain time-resolution limited by laser pulse width. It
%is also easy to detect the transmitted signal, but if one wants to
%measure very small differential transmission, careful attention
%has to be paid to minimize all the noise components. In the
%absence of any other noise, photon shot noise becomes the limiting
%factor.

\vspace{0.1in}

\bf 2. Raman Spectroscopy. \rm

In this form of pump-probe spectroscopy, the changes induced by
the pump are measured by the change in the Raman scattering signal
generated by the probe. In degenerate pump-probe Raman
spectroscopy, the two beam are derived from the same laser and are
generally cross-polarized to distinguish the Raman signals
generated by the pump and the probe. If the two lasers have
different wavelength, the signals can be distinguished spectrally.
The time-integrated Raman signal of the probe, or the total energy
of the Raman-scattered probe pulse, is measured as a function of
time delay between the pump and the probe. Both vibrational and
electronic Raman signals have been used in this form of
time-resolved Raman spectroscopy to monitor the dynamic of
photoexcited phonons and electronic excitations.

\vspace{0.1in}

\bf 3. Luminescence Spectroscopy \rm

Luminescence dynamics can be measured directly with fast detector
and fast electronics. This includes the use of fast photodiodes
which can provide time resolution in the picosecond for intense
pulses, but are not well-suited for low-level signal detection.
Therefore the correlation and upconversion spectroscopy are used
alternatively.

\vspace{0.1in}

\bf (a) Correlation Spectroscopy. \rm In this really form of
pump-probe technique, two pulse trains delayed by $\tau_{d}$
excite the semiconductor and the time-integrated luminescence
spectra are measured as a function of $\tau_{d}$. If the
luminescence signal is a nonlinear function of the excitation
density, then the signal with two excitation pulses is not simply
the sum of the signal with each beam exciting the sample
separately. In practice, one chops the two beams at two different
frequencies, and the nonlinear correlation signal is measured at
the sum frequency. The advantages of the technique are its
simplicity, and high time resolution limited only by the laser
pulse width. The disadvantage is the complexity in the
interpretation of the results.

\vspace{0.1in}

\bf (b) Upconversion Spectroscopy. \rm This technique provides a
means for measuring low-level luminescence signal with time
resolution limited by laser pulse width and has been aplied to a
wide variety of investigations in semiconductors. The ultrafast
laser is divided into two beams. The first part excites the
sample. The induced sample emission is collected in the usual
manner and focused on a nonlinear crystal. The second laser beam
is suitably delayed and also focused on the nonlinear crystal to
overlap the luminescence spot. When the laser beam and the
emission signal are present simultaneously in the nonlinear
crystal, frequency mixing occurs, which results in the generation
of an up-converted signal. For a given detection axis, the
up-converted frequency is determined both by the angle between two
incoming beams and the angle of the nonlinear crystal, and by the
optical frequencies of these beams. The nonlinear crystal thus
acts as an optical gate which is opened only when the laser pulse
is present in it. By varying the optical path of the laser pulse
which opens the gate, the photoluminescence signal is sampled at
different time delays. Figure 2 illustrates a schematic diagram of
luminescence upconversion spectroscopy.
\begin{figure}[htb]
\centerline{\psfig{figure=Fig3.eps}}

\caption{\em{Upconversion spectroscopy: (a) the principle, (b) the
temporal and spectral sampling of the luminescence signal, and (c)
an experimental setup \cite{ref2}.}}
\end{figure}

FOUR WAVE MIXING (FWM) SPECTROSCOPY

\vspace{0.1in}

\bf 1. Principle of the method. \rm

FWM experiments are generally used to measured the nonlinear
response of a semiconductor in terms of third-order polarization
induced by the optical fields. While the diffracted signal in FWM
experiment measures only the coherent response of the
semiconductor, the differential transmission or reflection signal
is determined by the coherent as well as the incoherent dynamics.

The four wave mixing process can be interpreted as an interaction
between four photons. A photon of frequency $\omega_{1}$ combines
with a photon of frequency $\omega_{2}$ to produce a photon of
frequency $\omega_{3}$ and another of frequency $\omega_{4}$. The
conservation of energy and momentum are
$\omega_{1}+\omega_{2}=\omega_{3}+\omega_{4}$ and
$\mathbf{k_{1}}+\mathbf{k_{2}}=\mathbf{k_{3}}+\mathbf{k_{4}}$
respectively. If all four waves are of the same frequency,
$\omega_{1}=\omega_{2}=\omega_{3}=\omega_{4}=\omega$, the process
is then called degenerate four wave mixing \cite{ref3}.

In the simplest form of FWM spectroscopy, the two-beam Degenerate
Four Wave Mixing (DFWM), the pump photons with wavevector
$\mathbf{k_{1}}$ generate a coherent polarization in the sample,
and the probe photons with wavevector $\mathbf{k_{2}}$ arrive at
the sample after a time delay $\tau_{d}$. If $\tau_{d}$ is smaller
than the dephasing time of polarization, an interference grating
is produced from which $\mathbf{k_{2}}$ can be self-diffracted
along the phase matched direction
$\mathbf{k_{d}}=2\mathbf{k_{2}}-\mathbf{k_{1}}$.

In a three-beam DFWM, the sample is excited by three beams
traveling along $\mathbf{k_{1}}$, $\mathbf{k_{2}}$ and
$\mathbf{k_{3}}$ arriving at the sample at times $t_{1}$, $t_{2}$
and $t_{3}$ respectively. $\tau_{21}$, $\tau_{31}$ and $\tau_{32}$
($\tau_{ij}-t_{i}-t_{j}$) define the delays between these beams.
The diffracted total energy along the phasematched direction
$\mathbf{k_{d}}=\mathbf{k_{3}}+\mathbf{k_{2}}-\mathbf{k_{1}}$ is
measured as a function of various time delays to obtain different
information about the system being investigated. Figure 3 shows
the schematic diagram for a three-beam FWM experiment.
\begin{figure}[htb]
\centerline{\psfig{figure=Fig5.eps}}

\caption{\em Schematic diagram for a three-beam FWM experiment.}
\end{figure}

\bf 2. FWM and third-order nonlinear effects. \rm

The nonlinear effect is characterized by a nonlinear relation
between polarization density of a dielectric medium and external
electric field. The relation can be customary written in the form

\vspace{0.05in}

\centerline{$P=\epsilon_0\chi E + 2dE^{2} + 4\chi^{(3)}E^{3}+...$}

\vspace{0.05in}

where \it P \em and \it E \em are the polarization and electric
field, $\chi$, \it d \em and $\chi^{(3)}$ are coefficients
describing the linear effect, the second- and third-order
nonlinear effects, respectively \cite{ref3}.

We now examine the case of two-beam DFWM experiments where one
measures the diffracted signal along $\mathbf{k_d}$. If the pulse
along $\mathbf{k_2}$ is delayed by time delay $\tau_d$ with
respect to the pulse along $\mathbf{k_1}$, then the electric field
can be expressed as

\vspace{0.1in}

\centerline{$\mathbf{E}\mathnormal{}(\mathbf{R}\mathnormal{},t)=
E_1A(\mathbf{R}\mathnormal{},t)exp[i(\mathbf{k_1.R}\mathnormal{}-\omega
t)]+E_2A(\mathbf{R}\mathnormal{},t-\tau_d)exp[i(\mathbf{k_2.R}\mathnormal{}-\omega
t)]$}

\vspace{0.1in}

Here $E_{1}$($E_{1}$) is the amplitude of the first(second)
electric field, $A(\mathbf{R}\mathnormal{,t)}$ is the electric
field pulse shape at $\mathbf{R}$, $\omega$ corresponds to the
peak of the mode-locked laser spectrum. Also we have ignored
polarization effects on dipole matrix and the relative phase,
absorbing 1/2 in the field amplitude and assumed that directions
of linear polarization of the first and second fields are equal
\cite{ref1}.

For a two-level system with the ground state with state vector
$|a\rangle$ and energy $E_{a}$ and an excited state with the state
vector $|b\rangle$ and energy $E_{b}$, by solving the optical
Bloch equations \cite{ref1,ref2}, we can obtain the third-order
polarization
\begin{eqnarray}
p^{(3)}_{221}=&-&2i(e|r_{ba}|/\hbar)^{3}exp(-gt)E_{1}E_{2}^2\int\limits_{-\infty}^{t}dt'''\int\limits_{-\infty}^{t'''}dt''\int\limits_{-\infty}^{t''}dt'\nonumber\\
&\times&\{exp(Gt''')exp[-g_r(t'''-t'')]A(\mathbf{R}\mathnormal{},t'''-\tau_d)\}\nonumber\\
&\times&\{A(\mathbf{R}\mathnormal{},t''-\tau_d)A(\mathbf{R}\mathnormal{},t')exp[-G*(t''-t')]\nonumber\\
&\qquad&+A(\mathbf{R}\mathnormal{},t'-\tau_d)A(\mathbf{R}\mathnormal{},t'')exp[-G(t''-t')]\}\nonumber
\end{eqnarray}
where \textbf{r} is the electron coordinate with respect to the
location of the nucleus at \textbf{R}, $G=g-i\omega$ with
$g=(i\Omega+1/T_2)$, $\Omega=(E_b-E_a)/\hbar$, $g_r=1/T_1$, $T_1$
is the lifetime of the state $b$ and $1/T_2$, the transverse
relaxation rate, is the sum of the recombination rate ($1/T_1$)
and pure dephasing rate (one can think of $T_2$ as the lifetime of
the coherent superposition state) \cite{ref1}.

The diffracted signal along $\mathbf{k_d}$ can be approximated by
the macroscopic polarization $P^{(3)}_{221}$. In practice, the
temporal evolution of the FWM (TR-FWM) signal in the
$\mathbf{k_d}$ direction

\vspace{0.1in}

\centerline{$S^{(3)}_{221}(t)=|P^{(3)}_{221}|^2$}

\vspace{0.1in}

and the time-integrated (TI-FWM) signal as a function of time
delay $\tau_d$

\vspace{0.1in}

\centerline{$I^{(3)}_{221}(\tau_d)=\int_{-\infty}^{\infty}S^{(3)}_{221}(t)dt$}

\vspace{0.1in}

\bf 3. Example of Application: Quantum Beats of Excitons \rm

Consider the case of a three-level system which has a common
ground state and two excited states that are close in energy
(Figure 4.a). If the system is excited resonantly with a short
laser pulse such that the spectral width of the laser is greater
than the energy separation between the two excited states
$|1\rangle$ and $|2\rangle$, then the laser pulse excites a
coherent superposition state, i.e. a linear superposition of the
two wavefunctions. This is a non-stationary state whose time
evolution is determined by the time-dependent Schrödinger
equation. Since the energies of the eigenstates comprising the
non-stationary state are different, the phase of the two
eigenstates will evolve at a different rate. It can be shown that
the spontaneous emission intensity from this system following the
pulsed excitation oscillates with a period given by $h/\Delta E$
where $\Delta E$ is the energy separation between levels
$|1\rangle$ and $|2\rangle$ and $h$ is the Planck's constant. This
is the phenomenon of quantum beats.
\begin{figure}[htb]
\centerline{\psfig{figure=Fig41.eps}}

\caption{\em (a) Schematic of a three-level system with two
closely-spaced excited states and a common state; (b) two
two-level systems with close transition energies but no common
state \cite{ref1}.}
\end{figure}
The term "quantum beats" has also been used to describe
interference phenomena which arise when the two transitions not
sharing a common ground or excited states are excited by a short
laser pulse whose spectrum is wider than the difference in
transition energies (Figure 4.b). Although such interference has
been considered as a special case of "quantum beats", there is no
wavefunction superposition or quantum interference. The following
part will present results obtained by several authors in studying
heavy hole-light hole (HH-LH) beats in semiconductor
heterostructures.

It is well known that removal of the translational symmetry in the
growth direction in quantum wells lifts the degeneracy between the
HH and the LH valence bands at $k=0$, and leads to well-separated
HH and LH excitons in quantum wells. This is an intrinsic
three-level system with a common ground state. Figure 5 shows
TI-FWM signals  from a 10 period 170/150 $\dot{A}$ $
GaAs/Al_{0.36}Ga_{0.64}As$ MQW sample at 5 K and a comparison of
the experiment data with a theoretical calculation. The laser is
$\approx 500$ fs pulse width and 4.2 meV spectral width. It can be
seen clearly the strong periodic modulation of the TI-FWM signal
when the laser is tuned to an energy between the HH and the LH
excitons. The period of the oscillations is $960\pm 40$ fs, in
good agreement with the period of 980 fs calculated from the 4.2
meV HH-LH exciton splitting observed in the PLE spectrum
\cite{ref1}.
\begin{figure}[htb]
\centerline{\psfig{figure=Fig6.eps}}

\caption{\em (a) TI-FWM signal from a 10 period 170/150 $\dot{A}$
$GaAs/Al_{0.36}Ga_{0.64}As$ MQW sample at 5 K and 4.2 meV;
dash-dot curve: 1.5265 eV excitation (exciting primary the HH
exciton) showing smooth decay; dash curve: 1.5310 eV, exciting
both the HH and the LH excitons showing the quantum beats; (b)
comparison with calculations based on a simple three-level model
(dash-dot line) \cite{ref1}.}
\end{figure}
\begin{thebibliography}{99}
\bibitem{ref1} J. Shah, \it Ultrafast Spectroscopy of
Semiconductors and Semiconductor Nanostructures \em , Springer,
Berlin, 1999.
\bibitem{ref2} C. Rullière, \it Femtosecond Laser Pulses:
Principles and Experiments \em , Springer, Berlin, 1998.
\bibitem{ref3} B. E. A. Saleh, M. C. Teich, \it Fundamentals of
Photonics \em , John Wiley, New York, 1991.

\end{thebibliography}
\end{document}
