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Coherent optical control of spin dynamics in semiconductor
quantum dots
G Slavcheva and O Hess
Advanced Technology Institute, School of Electronics and Physical Sciences,
University of Surrey, Guildford, GU2 7XH, Surrey, United Kingdom
[email protected]
Abstract. We develop a new general model for rigorous theoretical description of circularly
polarised ultrashort optical pulse interactions with the resonant nonlinearities in semiconductor
QDs embedded in optical waveguides and semiconductor microcavities. The method is based
on the self-consistent FDTD-solution of the vector Maxwell equations coupled via
macroscopic polarisation to the originally derived time-evolution equations of a discrete 4level quantum system in terms of the real pseudospin (coherence) vector exploiting the SU(4)
group formalism. Selective excitation of specific spin-states with predefined helicity of the optical
pulse and formation of polarised Self-Induced Transparency (SIT)-solitons in a specially prepared
degenerate four-level system is numerically demonstrated. The model is applied to the stimulated
optical dipole transitions of the trion state in a singly charged QD taking into account the spin
relaxation dynamics. Our theoretical and numerical approach yields the time evolution of the
spin population of the trion state which is in good agreement with the time-resolved polarised
photoluminescence experimental data.
1. Introduction
The all-optical spin orientation using circularly polarised optical pulses is a technique of key
importance for coherent generation and manipulation of spin-polarised states in semiconductor
nanostructures. Coherent-carrier control in semiconductor quantum dots (QDs) allows coherent
manipulation of the carrier wave functions on a time scale shorter than typical dephasing times.
Semiconductor QDs, or so-called artificial atoms have been shown to be excellent candidates for the
physical implementation of the objectives of the quantum computation. This is mainly due to the
similarities between QDs and atomic systems, such as the discrete-level electronic structure.
Coherent quantum control of the electron spin confined in singly charged semiconductor QDs has
recently attracted significant interest because it allows coherent preparation and detection of the single
electron spin states on a time scale shorter than typical spin decoherence times. Despite the recent progress
made [1], the coherent optical generation and read out of single electron spin states remains a challenging
task from experimental point of view. In this paper we address the problem by employing a novel
dynamical model of the ultrafast optical circularly polarised pulse interactions with the resonant
nonlinearities in a QD.
2. Theoretical background
The electromagnetic wave incident to a generic quantum (0-D or 2-D) system is propagating along the
quantization direction z and is elliptically polarised in a plane perpendicular to it. The optical wave
centre carrier frequency is tuned in resonance with the fundamental heavy-hole transition in a
QW/trion transition in a singly charged QD (Figure 1).
Figure 1 Schematic representation of the initial electron states in a single singly-charged quantum dot
(a,b) and the ground trion state created by  -(c) and +(d) light; (e) Energy-level diagram of a negatively
charged exciton (trion) state in a single quantum dot. The levels are labelled by the total angular
momentum projection Jz. The coherent optical transitions excited by  --and +- polarised light, the
spontaneous optical transitions and the spin-relaxation processes involved in the dynamics are designated
by arrows.
The dipole-allowed transitions correspond to total angular momentum projection difference Jz=1
and the fundamental energy gap is 0. The real and imaginary part of the Rabi frequencies associated with
the coherent optical transitions are given by:  x   E x ;
y 

 , where is the optical dipole
Ey

transition matrix element. The resonant nonlinearity is modelled by an ensemble of four-level systems
with density Na. The time-evolution of the quantum system in an external perturbation in terms of the
SU(4) Lie group [2] is given by:
S i
1
 f ijk j S k  S i  S ie 
t
Ti
(1)
where the system damping is accounted for by phenomenological non-uniform relaxation times Ti, f is
the fully antisymmetric tensor of the structure constants of SU(4) group, and the pseudospin S and the
torque vector  are expressed in terms of the λ-generators of SU(4) Lie algebra.
Assuming that the initial spin population resides on the lower-lying levels, the system Hamiltonian is given
by:
1


0
  x  i y 
0
0


2


1
    i 

0
0
0
x
y

Hˆ    2

1
0
0
0
  x  i y 

2


1
0
0
  x  i y 
0


2


(2)
Using the Hamiltonian a system describing the time-evolution of the 15-dimensional state vector S is
obtained. The vector Maxwell equations for the optical wave propagation [3] coupled via polarisation to the
pseudospin equations (1) are solved self-consistently in the time domain employing the FDTD technique [4].
3. Simulation results
The initial boundary value problem requires the knowledge of the whole time history of the initial
field along some characteristic, e.g. at z=0 (the left boundary of our simulation domain). The
circularly-polarised optical pulse is modelled by two orthogonal linearly polarised optical waves,
phase-shifted by /2:
Ex z  0, t   Eo sech 10  cos (ot )
E y z  0, t    E0 sech 10 sin (ot )
Ex z  0, t   Eo sech 10  cos (ot )
(3)
E y z  0, t   E0 sech 10 sin (ot )
 
 

 

where E0 is the initial field amplitude,   [t  T p / 2 ] / T p / 2 and Tp is the pulse duration. The
simulation domain is 150 m long with resonantly absorbing/amplifying four-level medium embedded
between two free-space regions each with length 7.5 m. The carrier frequency resonant with the
heavy-exciton dipole transition is taken to be 0=1.2558  1015 rad.s-1, corresponding to a wavelength
=1.5 m.
3.1. SIT in a four-level system
We shall focus on the high-intensity nonlinear regime of pulse propagation through an ensemble of 4-level
resonant dipoles. The SIT criterion is satisfied using an ultrashort pulse with duration Tp=100 fs and setting
the relaxation times T13=T14=T15=100 ps, T1=T6=T7=T12=100 ps, while setting the rest (describing crossed or
spin-flip transitions) to infinity, thereby reducing the system to a pair of two independent two-level systems.
We choose the initial pulse amplitude to correspond to a -pulse according to the Pulse Area Theorem, giving
E0=2.1093109 Vm-1. A circularly polarised -pulse excites (or de-excites) completely the two-level system
[3]. The simulation results for the ultrashort pulse propagation are shown in Figure 2 (a,b) for both
polarisations of the injected ultrashort pulse  - and +. Selective excitation of specific spin-polarised states is
numerically demonstrated depending on the helicity of the injected utrashort optical pulse.
Figure 2 Selective excitation of the system |1 |2 by  - optical pulse (a); selective excitation of the system
|3 |4 by + optical pulse (b)
Self-induced transparency effects and polarised soliton formation is shown for a four-level system
in Figure 3. The ultrashort pulse travels undistorted through the degenerate four-level medium driving
locally the population through full Rabi-flops.
Figure 3 A snapshot of a polarised soliton in a 4-level medium initially prepared in a state with population
residing in the lower-lying level |1 and in the upper level |4 (see Figure 1) at the simulation time t=200 fs.
3.2. Trion optically-induced spin dynamics
We apply the model to modulation-doped lens-shaped quantum dots with lateral dimensions largely
exceeding the height that is sufficiently general to represent a wide class of zero-dimensional systems. Due to
the quasi cylindrical symmetry, the energy level scheme of a single trion confined in such a dot has the
configuration shown in Figure 1. Intense resonant ultrafast optical pulse drives the transitions between the
electron and trion states. We should note that the time-dependence of the optically-induced coherent spin
generation and subsequent relaxation in a single quantum dot, averaged over a large number of successive
measurements is equivalent to the corresponding spin dynamics for an ensemble of quantum dots assumed in
the model.
The system under investigation is a GaAs/AlGaAs self-assembled modulation-doped MBE-grown QD
with 5 nm height [1] sandwiched between two 50 nm Al0.3Ga0.7As barriers with refractive indices nGaAs=3.63
and nAlGaAs=3.46 at the trion transition resonance wavelength. The circularly polarised pulse centre frequency
0= 2.393321015 rad.s-1 is tuned in resonance with the energy splitting between the ground electron and
excited trion state of ~1.58 eV [1], corresponding to a wavelength λ=787 nm and the pulse duration is Tp=1 ps
modulated by hyperbolic secant envelope (eq. 3) . Throughout the simulations the initial field amplitude is
varied from E0=5106 Vm-1 to E0=4107 Vm-1 and the trion dipole moment is 4.810-28 Cm. We shall be
interested in the low-temperature regime. Transitions between the lower-lying initial electron levels occur due
to the hyperfine interaction of the electron spin with the frozen random configuration of the nuclear spins of
the lattice ions [5] with spin decoherence times set to T3=T9=0.5 ns [6]. Transitions between the upper-lying
levels also occur due to the hole-spin relaxation in the trion state which is a two-phonon assisted process with
relaxation time T5=T11=sh ~ 1 µs [5]. The trion radiative decay (recombination) time is set
T13=T14=T15=r=100 ps. We assume that the crossed transitions are not allowed and set T2=T4=T8=T10=. The
trion spin dephasing time is taken to be T1=T6=T7=T12=0.5 ns [6]. Figure 4 (a,b) shows the time-dependence
of the spin population of all four states along with the E-field components time-evolution of the ultrashort  -circularly polarised pulse at two excitation field amplitudes.
Figure 4 Time evolution of the - optical pulse E-field components and the corresponding spin population of all
four states; 22 represents the trion |-3/2 state population for initial spin-down state proportional to the -polarised photoluminescence at two initial field amplitudes: (a) E0=5106 Vm-1; (b) E0=4107 Vm-1.
The initial spin population is assumed to reside in state |1 with spin-down orientation. The evolution
of the population 22 of state |2 describes the spin population of |-3/2  trion state excited by  - polarised
pulse, and therefore represents a measure of the intensity of the  --polarised photoluminescence, since
the rate of the  - photon emission is ~ 22/r. We demonstrate numerically the appearance of Rabi
oscillations at sufficiently high excitation intensities above threshold that suppress the electron spin
relaxation [5]. This in turn leads to longer decay times of the photoluminescence and therefore to the
possibility of detecting the spin state with greater accuracy. If the initial spin population resides in |3
with spin-up orientation, the optical excitation with the same --polarisation does not affect the second
system and the spin population of the |-3/2  trion state remains very close to zero, slowly accumulating
in time solely due to transitions between levels |1 |3 and |2 |4. This allows differentiation
between the two polarisations of the detected polarised photoluminescence and therefore a highprecision measurement of the spin state. The results are in good agreement with the time dependent
Faraday rotation experiments [7].
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