Robust manipulation of electron spin coherence in an ensemble of singly charged
quantum dots
A. Greilich, M. Wiemann, F. G. G. Hernandez † , D. R. Yakovlev § , I. A. Yugova ‡ , and M. Bayer
Experimentelle Physik II, Universität Dortmund, D-44221 Dortmund, Germany
A. Shabaev ⋆ and Al. L. Efros
Naval Research Laboratory, Washington, DC 20375, USA
D. Reuter and A. D. Wieck
Angewandte Festkörperphysik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
(Dated: November 4, 2018, robustcontrol-03-27-07-fin.tex)
Using the recently reported mode locking effect [1] we demonstrate a highly robust control of
electron spin coherence in an ensemble of (In,Ga)As quantum dots during the single spin coherence
time. The spin precession in a transverse magnetic field can be fully controlled up to 25 K by
the parameters of the exciting pulsed laser protocol such as the pulse train sequence, leading to
adjustable quantum beat bursts in Faraday rotation. Flipping of the electron spin precession phase
was demonstrated by inverting the polarization within a pulse doublet sequence.
PACS numbers: 72.25.Dc, 72.25.Rb, 78.47.+p, 78.55.Cr
The spin of an electron in a quantum dot (QD) is an
attractive quantum bit candidate [2, 3, 4, 5] due to its
favorable coherence properties [1, 6, 7, 8]. As the inter-
action strength is rather small for direct spin manipula-
tion, the idea to swap spin into charge has been furbished
[6, 9, 10]. For example, the electron may be converted
into a charged exciton by optical injection of an electron-
hole pair [10], depending on the residual electron’s spin
orientation, leading to distinctive polarization selection
rules.
The fundamental quantity regarding spin coherence is
the transverse relaxation time T2 . In a QD ensemble,
this time is masked by dephasing, mostly caused by dot-
to-dot variations of the spin dynamics. The dephasing
time does not exceed 10 ns, much shorter than T2 . This
leads to the general believe that manipulations ought to
-1 0 1 2 3 4 5 -1 0 1 2 3 4 5
= 1.86 ns
3.26 ns
3.66 ns
3.76 ns
3.86 ns
= 4.26 ns
B = 6 T, T = 6 K
4.92 ns
5.22 ns
5.42 ns
5.62 ns
5.92 ns
Time (ns)
FIG. 1: Faraday rotation traces measured as function of delay
between probe and first pump pulse at time zero. A second
pump pulse was applied, delayed relative to the first one by
TD , indicated at each trace. The top left trace gives the FR
without second pump.
be performed on a single spin. Measurement of a single
electron spin polarization, however, also results in de-
phasing due to temporal sampling of varying nuclear spin
configurations [11, 12], as statistically significant mea-
surements on a single QD may require multiple repetition
of the experiment. The dephasing can be overcome by
spin-echo techniques, which give a single electron spin co-
herence time on the scale of micro-seconds [8]. This long
coherence time derived by spin-echo is result of a refo-
cusing of the electron spin and possibly the nuclear spin
configuration [11], and it is viewed as an upper bound on
the free-induction decay of spin coherence [11, 13].
Recently, however, we have shown that mode locking
of electron spin coherence allows one to overcome the en-
semble dephasing [14] and to measure the single electron
spin relaxation time T2 without applying spin-echo re-
focusing [1]. For monitoring the coherence, pump-probe
Faraday rotation (FR) measurements [15] on a QD en-
semble were used: after optical alignment of the spins
normal to an external magnetic field the electron spins
precess about this field. Due to precession frequency
variations the ensemble phase coherence is quickly lost.
However, a periodic train of circularly polarized pulses
emitted by a mode-locked laser synchronizes those spin
precession modes, for which the precession frequency is
a multiple of the laser repetition rate. This synchroniza-
tion leads to constructive interference (CI) of these modes
in the ensemble spin polarization before arrival of each
pump pulse (see Fig. 1, upper left trace). The limit for
spin mode locking is set by the single electron spin co-
herence time which can last up to a few microseconds [1]
reaching the low bound on echo-like decays [16].
Here we develop a detailed understanding of the de-
gree of control which can be reached for the electron spin
coherence in an ensemble of singly charged QDs by ex-
ploiting the mode locking. For this purpose trains of
excitation pump pulse doublets were designed to vary
http://arxiv.org/abs/0704.0177v1
the phase synchronization condition (PSC) for electron
spin precession frequencies. The PSC selects a QD sub-
set, whose contribution to the ensemble spin polarization
shows a well controlled phase recovery. Variation of the
pulse separation results in tunable patterns of quantum
oscillation bursts in time-resolved FR, in good agreement
with our calculation, which rely on a newly developed
theoretical model. This tailoring of electron spin coher-
ence is very robust, as the spin mode locking is stable up
to 25 K. For higher temperatures the coherence ampli-
tude decreases due to phonon-assisted scattering of holes
during the laser pulse excitation by which the spin co-
herence is created.
The studied self-assembled (In,Ga)As/GaAs QDs were
fabricated by molecular beam epitaxy on a (001)-oriented
GaAs substrate. The sample contains 20 QD layers with
a layer dot density of about 10 10 cm −2 , separated by 60
nm wide barriers [17]. For average occupation by a sin-
gle electron per dot, the structures were n -modulation
doped 20 nm below each layer with a Si-dopant density
matching roughly the dot density. The sample was held
in the insert of an optical magneto-cryostat, allowing
temperature variation from T = 6 to 50 K.
FR with picosecond time resolution was used for study-
ing the spin dynamics: Thereby spin polarization along
the growth direction ( z -axis) is generated by a circu-
larly polarized pump pulse hitting the sample along z ,
and its precession in a transverse magnetic field B ≤ 7 T
along the x -axis is tested by the rotation of the linear
polarization of a probe pulse. For optical excitation, a Ti-
sapphire laser was used emitting pulses with a duration
of ∼ 1.5 ps (full width at half maximum of ∼ 1 meV) at
75.6 MHz repetition rate (corresponding to a repetition
period TR = 13.2 ns). The laser energy was tuned into
resonance with the QD ground state transition and the
laser pulses were split into pump and probe. The pump
beam was split further into two pulses with variable de-
lay TD in between. The circular polarization of the two
pumps could be controlled individually. For detecting
the rotation angle of the probe beam linear polarization,
a homodyne technique was used.
Figure 1 shows FR traces excited by the two-pulse
train with a repetition period TR = 13.2 ns, in which
both pulses have the same intensity and polarization,
and the delay between these pulses TD was varied be-
tween ∼ TR/7 and ∼ TR/2 . The FR pattern varies
strongly for the case when the delay time TD is commen-
surate with the repetition period TR : TD = TR/i with
i = 2, 3, ... , and for the case TD 6= TR/i . For commen-
surability TD = TR/i , the FR signal shows strong peri-
odic bursts of quantum oscillations only at times equal
to multiples of TD , as seen in the left panel of Fig. 1 for
TD = 1.86 ns≈ TR/7 . Commensurability is also given to
a good approximation for delays TD = TR/4 ≈ 3.26 ns
and TD = TR/3 ≈ 4.26 ns.
For incommensurability of TD and TR , TD 6= TR/i ,
the FR signal shows bursts of quantum oscillations be-
tween the two pulses of each pump doublet, in addition
to the bursts outside of the doublet. For example, one
can see a single burst in the mid between the pumps
for TD = 3.76 and 5.22 ns. Two bursts, each equidis-
tant from the closest pump and also equidistant from
one another, appear at TD = 4.92 and 5.62 ns. Three
equidistant bursts occur at TD = 5.92 ns. Note also that
the FR amplitude before the second pump arrival is al-
ways significantly larger than before the first pump for
any TD .
Although the time dependencies of the FR signal look
very different for commensurate and incommensurate
TD and TR , in both cases they can be fully controlled
by designing the synchronization of electron spin pre-
cession modes in order to reach CI of their contribu-
tions to the FR signal [1]. A train of circularly polar-
ized pump pulse singlets synchronizes those spin preces-
sions for which the precession frequency satisfies the PSC
[1, 18]: ωe = 2πN/TR . Then the electron spin under-
goes an integer number, N ≫ 1 , of full 2π rotations in
the interval TR between the pump pulses.
For a train of pump pulse doublets the PSC has to be
extended to account for the intervals TD and TR − TD
in the laser excitation protocol
ωe = 2πNK/TD = 2πNL/(TR − TD) , (1)
where K and L are integers. On first glance this con-
dition imposes severe limitations on the TD values, for
which synchronization is obtained:
TD = [K/(K + L)]TR , (2)
which for TD < TR/2 leads to K < L . When Eq. (2)
is satisfied, the contribution of synchronized precession
modes to the average electron spin polarization Sz(t) is
proportional to −0.5 cos[N(2πKt/TD)] . Summing over
all relevant oscillations leads to CI of their contributions
with a period TD/K in time [1]. The rest of QDs does
not contribute to Sz(t) at times longer than the ensem-
ble dephasing time. The PSC Eq. (1) explains the posi-
tion of all bursts in the FR signal for commensurate and
incommensurate ratios of TD and TR . For commensu-
rability, K ≡ 1 and TD = TR/(1 + L) according to Eq.
(2). In this case CIs should occur with period TD as
seen in Fig. 1 for TD = 1.86 ns (L = 6 ).
For incommensurability of TD and TR the number of
FR bursts between the two pulses within a pump doublet
and the delays at which they appear can be tailored.
There should be just one burst between the pulses, when
K ≡ 2 , because then the CI must have a period TD/2 .
A single burst is seen in Fig. 1 for TD = 3.76 and
5.22ns. The corresponding ratios TD/TR are 0.285 and
0.395, respectively. At the same time Eq.(2) gives a ratio
TD/TR = 2/(L+ 2) , which is equal to 0.285 and 0.4 for
L=5 and 3, respectively, in good accord with experiment.
Next, two FR bursts are seen for TD = 4.92 and
5.62ns, corresponding to TD/TR ≈ 0.372 and 0.426. The
corresponding CI period TD/3 is reached for K ≡ 3 .
Then from Eq.(2) TD/TR = 3/(L + 3) , giving 0.375
-1 0 1 2 3 4 5 6
305 310 315 305 310 315
TD=2TR/7  
Time (ns)
TD=TR/3
t = TD
t = 0
TD=TR/3
 Precession frequency (GHz)
t = 0
t = TD
TD=2TR/7
FIG. 2: (a,b): Spectra of electron spin precession modes,
−Sz(t) , which are phase synchronized by the two-pulse train
calculated for TD = TR/3 and TD = 2TR/7 at the moments
of first ( t = 0 ) and second ( t = TD ) pulse arrival (red).
Single-pump spectra are shown in blue. (c): FR traces cal-
culated for two ratios of TD/TR . Laser pulse area Θ = π .
TR = 13.2 ns. Electron g -factor | ge |= 0.57 and its disper-
sion ∆g = 0.005 . B = 6 T.
and 0.429 for L = 5 and 4, respectively. Finally, the
FR signal with TD = 5.92 ns ( TD/TR ≈ 0.448) shows
three FR bursts between the two pumps. The CI period
TD/4 is obtained for K ≡ 4 , for which Eq.(2) gives
TD/TR = 4/(L + 4) ≈ 0.444 with L = 5 . Obviously
good general agreement between experiment and theory
is established, highlighting the high flexibility of the laser
protocol. In turn, this understanding can be used to in-
duce FR bursts at wanted delays TD/K , so that at these
times further coherent manipulation of all electron spins
involved in the burst is facilitated.
However, the question arises how accurate condition
Eq. (2) for the TD/TR ratio must be fulfilled to reach
phase synchronization. Formally, one can find for any
arbitrary TD/TR large K and L values such that Eq.
(2) is satisfied with high accuracy. But the above analysis
shows, that only the smallest of all available L leads to
PSC matching. Experimentally, the facilities to address
this point are limited, as the largest TD for which FR
signal can be measured are delays around 5 ns between
the two pumps. For larger delays the FR bursts shift out
of the scanning range. For short TD , on the other hand,
the bursts are overlapping with the FR signal from the
pump pulses.
To answer this question, we have modeled the FR sig-
nal for commensurate and incommensurate ratios of TD
and TR . Figure 2 shows the results together with spec-
tra of synchronized spin precession modes (SSPM) at the
moment of the first and second pump pulse arrival. The
SSPM were calculated similar those induced by a single
pulse train [1]. Figure 2(a) gives the SSPM for com-
mensurate TD = TR/3 superimposed on the SSPM cre-
ated by a single pulse train with the same TR . Panel
(c) shows the FR signal created by such a two pulse
train. The SSPM for the considered strong excitation
are considerably broadened and contain modes for which
ωe = 2πM/TD = 2π3M/TR with integer M , which co-
incide with each third mode created by a single pulse
train. However, the SSPM given by ωe = 2πN/TR ,
which do not satisfy the PSC for a two pulse train, are
not completely suppressed, because the train synchro-
nizes the electron spin precession in some frequency range
around the PSC. One sees also, that at t = 0 the two
pulse train leads to a significant alignment of electron
spins opposite to the direction of spins satisfying the
PSC. This ”negative” alignment decreases the CI mag-
nitude and therefore the FR signal before the first pulse
arrival, and is also responsible for a significantly larger
magnitude of the FR signal before the second pulse ar-
rival [see Figs. 1 and 2 (c)].
For incommensurate ratios of TD and TR the SSPM
become much more complex. Still we are able to recover
the modes which satisfy the PSC at the pulse arrival
times. In Fig. 2 (b) we show the SSPM at t = 0 and t =
TD for TD = 2TR/7 (K = 2 , L = 5 ), where the arrows
indicate the frequencies which satisfy the PSC for the
two pulse train. Only a small number of such modes fall
within the average distribution of electron spin precession
modes, because the distance between the PSC modes is
proportional to 2πK/TD = 2π(K +L)/TR . The diluted
spectra of PSC modes for incommensurability decrease
the magnitude of the FR bursts between the pump pulses,
in accord with experiment. This shows, that although
any ratio of TD/TR can be satisfied by large K and
L , the FR signal between the pulses should be negligibly
small in this case. Consequently, not any ratio of TD/TR
leads to pronounced FR bursts.
To obtain further insight into the tailoring of electron
spin coherence, which can be reached by a two-pulse
train, we have turned from co- to counter-circularly po-
larized pumps. The delay between pumps TD was fixed
at TR/6 ≈ 2.2 ns. The time dependencies of the corre-
sponding FR signals are similar, as shown in Fig. 3. Be-
sides the two FR bursts directly connected to the pump
pulses, one sees a burst +1 due to CI of spin synchro-
nized modes. The insets in Fig. 3 (a) show closeups of
the different FR bursts. The sign, κ , of the FR am-
plitude for the counter-circular configuration undergoes
2TD -periodic changes in time relative to the co-circular
case, as seen in Fig. 3 (b), which demonstrates optical
switching of the electron spin precession phase by π in
an ensemble of QDs.
The observed effect of sign reversal is well described
by our model. Let us consider first a two-pulse train
with delay time TD = TR/2 for which the two pumps
-1 0 1 2 3 4 5
0 1 2 3 4 5
 co, 
 counter, 
 Time (ns)
pump 1 pump 2 +1 burst
- (b)  
 Time (ns)
B = 3 T T = 6 K (c)
FIG. 3: (a): Faraday rotation traces in the co-circularly (blue
traces) or counter-circularly (red traces) polarized two pump
pulse experiments measured for TD = 2.2 ns and B = 6T.
Insets give close-ups showing the relative sign, κ , of the FR
amplitude between the two traces. κ is plotted in (b) vs
time. (c): Effect of temperature on the FR amplitude in two-
pump-pulse experiment. TD = 1.88 ns.
are counter-circularly polarized. In this case an electron
spin can be synchronized only if at the moment of pulse
arrival it has an orientation opposite to the orientation
at the previous pulse. This leads to the PSC ωe =
2π(N + 1/2)/TD . The contribution of such precession
modes to the electron spin polarization is proportional
to cos[2π(N+1/2)t/TD)] = cos(2πNt/TD) cos(πt/TD)−
sin(2πNt/TD) sin(πt/TD) . Summing these contribu-
tions, only the first term gives a CI, whose modulus has
period TD , while the sign of cos(πt/TD) changes with
period 2TD . Only each third of the precession frequen-
cies can be synchronized by a counter-circularly polarized
two pulse train when the delay time is TR/6 as in our ex-
periment. However, the corresponding PSC has the same
dependence on TD . The CI modulus also has period TD
and its sign changes with period 2TD . The relative sign
of the FR amplitude for the counter- and co-circularly
case, κ = sgn{cos[πt/TD]} , is in accord with the exper-
imental dependence in Fig. 3 (b).
The CIs of the electron spin contributions can be seen
only as long as the coherence of the electron spins is main-
tained. In this respect the temperature stability of the
CI is especially important. Fig. 3 (c) shows FR traces
in a two-pump-pulse configuration with TD = 1.88 ns at
different temperatures. For both positive and negative
delays, the FR amplitude at a fixed delay is about con-
stant for temperatures up to 25 K, irrespective of slight
variations which might arise from changes in the phase
synchronization of QD subsets. Above 30 K a sharp drop
occurs, which can be explained by thermally activated
destruction of the spin coherence.
The electron spin coherence in charged QDs is initi-
ated by generation of a superposition of an electron and
a charged exciton state by resonant pump pulses [17, 19].
The simultaneous decrease of the FR magnitude before
each pump pulse and afterwards (when the CI signal is
controlled by the excitation pulse) suggests that the co-
herence at elevated temperatures is lost already during its
generation. The 30K temperature threshold corresponds
to an activation energy of ∼ 2.5 meV. This energy may
be assigned only to the splitting between the two lowest
confined hole levels, because the electron level splitting
dominates the 20 meV splitting between p- and s-shell
emission in photoluminescence and is much larger than
2.5 meV. The decoherence of the hole spin results from
two phonon scattering, which is thermally activated and
should occur on a sub-picosecond time scale, i.e. within
the laser pulse [20]. The fast decoherence of the hole spin
at T > 30 K suppresses formation of the electron-trion
superposition state. ps-pulses as used here are therefore
not sufficiently short for initialization of the superposi-
tion and creation of a long-lived electron spin coherence.
In summary, we have demonstrated that the mode-
locking effect allows a far-reaching control of electron
spin coherence in QD ensembles during the spin coher-
ence time of microseconds [1]. Two-pulse train mode-
locking selects QD subsets which give a non-dephasing
contribution to the ensemble spin precession. The tech-
nique shows remarkable stability with respect to temper-
ature increase up to 25 K, a property which is important
for utilizing it in quantum information processing. The
robustness of this control technique is provided by the
dispersion of the spin precession frequencies in the QD
ensemble.
Acknowledgments. This work was supported by the
BMBF program nanoquit, the DARPA program QuIST,
the ONR, the DFG (FOR485) and FAPESP.
[ † ] on leave from the Instituto de Fisica Gleb Wataghin,
Campinas, SP , Brazil.
[ § ] also at Ioffe Physico-Technical Institute, 194021, St. Pe-
tersburg, Russia.
[ ‡ ] also at Institute of Physics, St. Petersburg State Univer-
sity, 198504, St. Petersburg, Russia.
[ ⋆ ] also at School of Computational Sciences, George Mason
University, Fairfax VA 22030.
[1] A. Greilich et al., Science 313, 341 (2006).
[2] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120
(1998).
[3] A. Imamoglu et al., Phys. Rev. Lett. 83, 4204 (1999).
[4] Semiconductor Spintronics and Quantum Computation,
ed. by D. D. Awschalom, D. Loss, and N. Samarth
(Springer-Verlag, Heidelberg 2002).
[5] S. M. Clark et al., cond-mat/0610152.
[6] J. M. Elzerman et al., Nature 430, 431 (2004).
[7] M. Kroutvar et al., Nature 432, 81 (2004).
[8] J. R. Petta et al., Science 309, 2180 (2005).
[9] see, for example, R. Hanson et al., Phys. Rev. Lett. 94,
196802 (2005).
[10] see, for example, T. Calarco et al., Phys. Rev. A 68,
012310 (2003); P. Chen et al., Phys. Rev. B 69, 075320
(2004).
[11] W. Yao, R. Liu, and L. J. Sham, Phys. Rev. B 74, 195301
(2006).
[12] R. Liu, S. E. Economou, L. J. Sham, and D. G. Steel,
Phys. Rev. B 75, 085322 (2007).
[13] W. A. Coish et al., Phys. Stat. Sol. (b) 243, 3658 (2006).
[14] For a general treatment on suppression of phase noise
see A. G. Kofman and G. Kurizki, Phys. Rev. Lett. 93,
130406 (2004).
[15] J. M. Kikkawa and D. D. Awschalom, Science 287, 473
(2000).
[16] R. Hanson et al., cond-mat/0610433.
[17] A. Greilich et al., Phys. Rev. Lett. 95, 227401 (2006).
[18] A. Shabaev et al., Phys. Rev. B 68, 201305(R) (2003).
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[20] T. Takagahara, Phys. Rev. B 62, 16840 (2000).
http://arxiv.org/abs/cond-mat/0610152
http://arxiv.org/abs/cond-mat/0610433

  Using the recently reported mode locking effect we demonstrate a highly
robust control of electron spin coherence in an ensemble of (In,Ga)As quantum
dots during the single spin coherence time. The spin precession in a transverse
magnetic field can be fully controlled up to 25 K by the parameters of the
exciting pulsed laser protocol such as the pulse train sequence, leading to
adjustable quantum beat bursts in Faraday rotation. Flipping of the electron
spin precession phase was demonstrated by inverting the polarization within a
pulse doublet sequence.

Robust manipulation of electron spin coherence in an ensemble of singly charged
quantum dots
A. Greilich, M. Wiemann, F. G. G. Hernandez † , D. R. Yakovlev § , I. A. Yugova ‡ , and M. Bayer
Experimentelle Physik II, Universität Dortmund, D-44221 Dortmund, Germany
A. Shabaev ⋆ and Al. L. Efros
Naval Research Laboratory, Washington, DC 20375, USA
D. Reuter and A. D. Wieck
Angewandte Festkörperphysik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
(Dated: November 4, 2018, robustcontrol-03-27-07-fin.tex)
Using the recently reported mode locking effect [1] we demonstrate a highly robust control of
electron spin coherence in an ensemble of (In,Ga)As quantum dots during the single spin coherence
time. The spin precession in a transverse magnetic field can be fully controlled up to 25 K by
the parameters of the exciting pulsed laser protocol such as the pulse train sequence, leading to
adjustable quantum beat bursts in Faraday rotation. Flipping of the electron spin precession phase
was demonstrated by inverting the polarization within a pulse doublet sequence.
PACS numbers: 72.25.Dc, 72.25.Rb, 78.47.+p, 78.55.Cr
The spin of an electron in a quantum dot (QD) is an
attractive quantum bit candidate [2, 3, 4, 5] due to its
favorable coherence properties [1, 6, 7, 8]. As the inter-
action strength is rather small for direct spin manipula-
tion, the idea to swap spin into charge has been furbished
[6, 9, 10]. For example, the electron may be converted
into a charged exciton by optical injection of an electron-
hole pair [10], depending on the residual electron’s spin
orientation, leading to distinctive polarization selection
rules.
The fundamental quantity regarding spin coherence is
the transverse relaxation time T2 . In a QD ensemble,
this time is masked by dephasing, mostly caused by dot-
to-dot variations of the spin dynamics. The dephasing
time does not exceed 10 ns, much shorter than T2 . This
leads to the general believe that manipulations ought to
-1 0 1 2 3 4 5 -1 0 1 2 3 4 5
= 1.86 ns
3.26 ns
3.66 ns
3.76 ns
3.86 ns
= 4.26 ns
B = 6 T, T = 6 K
4.92 ns
5.22 ns
5.42 ns
5.62 ns
5.92 ns
Time (ns)
FIG. 1: Faraday rotation traces measured as function of delay
between probe and first pump pulse at time zero. A second
pump pulse was applied, delayed relative to the first one by
TD , indicated at each trace. The top left trace gives the FR
without second pump.
be performed on a single spin. Measurement of a single
electron spin polarization, however, also results in de-
phasing due to temporal sampling of varying nuclear spin
configurations [11, 12], as statistically significant mea-
surements on a single QD may require multiple repetition
of the experiment. The dephasing can be overcome by
spin-echo techniques, which give a single electron spin co-
herence time on the scale of micro-seconds [8]. This long
coherence time derived by spin-echo is result of a refo-
cusing of the electron spin and possibly the nuclear spin
configuration [11], and it is viewed as an upper bound on
the free-induction decay of spin coherence [11, 13].
Recently, however, we have shown that mode locking
of electron spin coherence allows one to overcome the en-
semble dephasing [14] and to measure the single electron
spin relaxation time T2 without applying spin-echo re-
focusing [1]. For monitoring the coherence, pump-probe
Faraday rotation (FR) measurements [15] on a QD en-
semble were used: after optical alignment of the spins
normal to an external magnetic field the electron spins
precess about this field. Due to precession frequency
variations the ensemble phase coherence is quickly lost.
However, a periodic train of circularly polarized pulses
emitted by a mode-locked laser synchronizes those spin
precession modes, for which the precession frequency is
a multiple of the laser repetition rate. This synchroniza-
tion leads to constructive interference (CI) of these modes
in the ensemble spin polarization before arrival of each
pump pulse (see Fig. 1, upper left trace). The limit for
spin mode locking is set by the single electron spin co-
herence time which can last up to a few microseconds [1]
reaching the low bound on echo-like decays [16].
Here we develop a detailed understanding of the de-
gree of control which can be reached for the electron spin
coherence in an ensemble of singly charged QDs by ex-
ploiting the mode locking. For this purpose trains of
excitation pump pulse doublets were designed to vary
http://arxiv.org/abs/0704.0177v1
the phase synchronization condition (PSC) for electron
spin precession frequencies. The PSC selects a QD sub-
set, whose contribution to the ensemble spin polarization
shows a well controlled phase recovery. Variation of the
pulse separation results in tunable patterns of quantum
oscillation bursts in time-resolved FR, in good agreement
with our calculation, which rely on a newly developed
theoretical model. This tailoring of electron spin coher-
ence is very robust, as the spin mode locking is stable up
to 25 K. For higher temperatures the coherence ampli-
tude decreases due to phonon-assisted scattering of holes
during the laser pulse excitation by which the spin co-
herence is created.
The studied self-assembled (In,Ga)As/GaAs QDs were
fabricated by molecular beam epitaxy on a (001)-oriented
GaAs substrate. The sample contains 20 QD layers with
a layer dot density of about 10 10 cm −2 , separated by 60
nm wide barriers [17]. For average occupation by a sin-
gle electron per dot, the structures were n -modulation
doped 20 nm below each layer with a Si-dopant density
matching roughly the dot density. The sample was held
in the insert of an optical magneto-cryostat, allowing
temperature variation from T = 6 to 50 K.
FR with picosecond time resolution was used for study-
ing the spin dynamics: Thereby spin polarization along
the growth direction ( z -axis) is generated by a circu-
larly polarized pump pulse hitting the sample along z ,
and its precession in a transverse magnetic field B ≤ 7 T
along the x -axis is tested by the rotation of the linear
polarization of a probe pulse. For optical excitation, a Ti-
sapphire laser was used emitting pulses with a duration
of ∼ 1.5 ps (full width at half maximum of ∼ 1 meV) at
75.6 MHz repetition rate (corresponding to a repetition
period TR = 13.2 ns). The laser energy was tuned into
resonance with the QD ground state transition and the
laser pulses were split into pump and probe. The pump
beam was split further into two pulses with variable de-
lay TD in between. The circular polarization of the two
pumps could be controlled individually. For detecting
the rotation angle of the probe beam linear polarization,
a homodyne technique was used.
Figure 1 shows FR traces excited by the two-pulse
train with a repetition period TR = 13.2 ns, in which
both pulses have the same intensity and polarization,
and the delay between these pulses TD was varied be-
tween ∼ TR/7 and ∼ TR/2 . The FR pattern varies
strongly for the case when the delay time TD is commen-
surate with the repetition period TR : TD = TR/i with
i = 2, 3, ... , and for the case TD 6= TR/i . For commen-
surability TD = TR/i , the FR signal shows strong peri-
odic bursts of quantum oscillations only at times equal
to multiples of TD , as seen in the left panel of Fig. 1 for
TD = 1.86 ns≈ TR/7 . Commensurability is also given to
a good approximation for delays TD = TR/4 ≈ 3.26 ns
and TD = TR/3 ≈ 4.26 ns.
For incommensurability of TD and TR , TD 6= TR/i ,
the FR signal shows bursts of quantum oscillations be-
tween the two pulses of each pump doublet, in addition
to the bursts outside of the doublet. For example, one
can see a single burst in the mid between the pumps
for TD = 3.76 and 5.22 ns. Two bursts, each equidis-
tant from the closest pump and also equidistant from
one another, appear at TD = 4.92 and 5.62 ns. Three
equidistant bursts occur at TD = 5.92 ns. Note also that
the FR amplitude before the second pump arrival is al-
ways significantly larger than before the first pump for
any TD .
Although the time dependencies of the FR signal look
very different for commensurate and incommensurate
TD and TR , in both cases they can be fully controlled
by designing the synchronization of electron spin pre-
cession modes in order to reach CI of their contribu-
tions to the FR signal [1]. A train of circularly polar-
ized pump pulse singlets synchronizes those spin preces-
sions for which the precession frequency satisfies the PSC
[1, 18]: ωe = 2πN/TR . Then the electron spin under-
goes an integer number, N ≫ 1 , of full 2π rotations in
the interval TR between the pump pulses.
For a train of pump pulse doublets the PSC has to be
extended to account for the intervals TD and TR − TD
in the laser excitation protocol
ωe = 2πNK/TD = 2πNL/(TR − TD) , (1)
where K and L are integers. On first glance this con-
dition imposes severe limitations on the TD values, for
which synchronization is obtained:
TD = [K/(K + L)]TR , (2)
which for TD < TR/2 leads to K < L . When Eq. (2)
is satisfied, the contribution of synchronized precession
modes to the average electron spin polarization Sz(t) is
proportional to −0.5 cos[N(2πKt/TD)] . Summing over
all relevant oscillations leads to CI of their contributions
with a period TD/K in time [1]. The rest of QDs does
not contribute to Sz(t) at times longer than the ensem-
ble dephasing time. The PSC Eq. (1) explains the posi-
tion of all bursts in the FR signal for commensurate and
incommensurate ratios of TD and TR . For commensu-
rability, K ≡ 1 and TD = TR/(1 + L) according to Eq.
(2). In this case CIs should occur with period TD as
seen in Fig. 1 for TD = 1.86 ns (L = 6 ).
For incommensurability of TD and TR the number of
FR bursts between the two pulses within a pump doublet
and the delays at which they appear can be tailored.
There should be just one burst between the pulses, when
K ≡ 2 , because then the CI must have a period TD/2 .
A single burst is seen in Fig. 1 for TD = 3.76 and
5.22ns. The corresponding ratios TD/TR are 0.285 and
0.395, respectively. At the same time Eq.(2) gives a ratio
TD/TR = 2/(L+ 2) , which is equal to 0.285 and 0.4 for
L=5 and 3, respectively, in good accord with experiment.
Next, two FR bursts are seen for TD = 4.92 and
5.62ns, corresponding to TD/TR ≈ 0.372 and 0.426. The
corresponding CI period TD/3 is reached for K ≡ 3 .
Then from Eq.(2) TD/TR = 3/(L + 3) , giving 0.375
-1 0 1 2 3 4 5 6
305 310 315 305 310 315
TD=2TR/7  
Time (ns)
TD=TR/3
t = TD
t = 0
TD=TR/3
 Precession frequency (GHz)
t = 0
t = TD
TD=2TR/7
FIG. 2: (a,b): Spectra of electron spin precession modes,
−Sz(t) , which are phase synchronized by the two-pulse train
calculated for TD = TR/3 and TD = 2TR/7 at the moments
of first ( t = 0 ) and second ( t = TD ) pulse arrival (red).
Single-pump spectra are shown in blue. (c): FR traces cal-
culated for two ratios of TD/TR . Laser pulse area Θ = π .
TR = 13.2 ns. Electron g -factor | ge |= 0.57 and its disper-
sion ∆g = 0.005 . B = 6 T.
and 0.429 for L = 5 and 4, respectively. Finally, the
FR signal with TD = 5.92 ns ( TD/TR ≈ 0.448) shows
three FR bursts between the two pumps. The CI period
TD/4 is obtained for K ≡ 4 , for which Eq.(2) gives
TD/TR = 4/(L + 4) ≈ 0.444 with L = 5 . Obviously
good general agreement between experiment and theory
is established, highlighting the high flexibility of the laser
protocol. In turn, this understanding can be used to in-
duce FR bursts at wanted delays TD/K , so that at these
times further coherent manipulation of all electron spins
involved in the burst is facilitated.
However, the question arises how accurate condition
Eq. (2) for the TD/TR ratio must be fulfilled to reach
phase synchronization. Formally, one can find for any
arbitrary TD/TR large K and L values such that Eq.
(2) is satisfied with high accuracy. But the above analysis
shows, that only the smallest of all available L leads to
PSC matching. Experimentally, the facilities to address
this point are limited, as the largest TD for which FR
signal can be measured are delays around 5 ns between
the two pumps. For larger delays the FR bursts shift out
of the scanning range. For short TD , on the other hand,
the bursts are overlapping with the FR signal from the
pump pulses.
To answer this question, we have modeled the FR sig-
nal for commensurate and incommensurate ratios of TD
and TR . Figure 2 shows the results together with spec-
tra of synchronized spin precession modes (SSPM) at the
moment of the first and second pump pulse arrival. The
SSPM were calculated similar those induced by a single
pulse train [1]. Figure 2(a) gives the SSPM for com-
mensurate TD = TR/3 superimposed on the SSPM cre-
ated by a single pulse train with the same TR . Panel
(c) shows the FR signal created by such a two pulse
train. The SSPM for the considered strong excitation
are considerably broadened and contain modes for which
ωe = 2πM/TD = 2π3M/TR with integer M , which co-
incide with each third mode created by a single pulse
train. However, the SSPM given by ωe = 2πN/TR ,
which do not satisfy the PSC for a two pulse train, are
not completely suppressed, because the train synchro-
nizes the electron spin precession in some frequency range
around the PSC. One sees also, that at t = 0 the two
pulse train leads to a significant alignment of electron
spins opposite to the direction of spins satisfying the
PSC. This ”negative” alignment decreases the CI mag-
nitude and therefore the FR signal before the first pulse
arrival, and is also responsible for a significantly larger
magnitude of the FR signal before the second pulse ar-
rival [see Figs. 1 and 2 (c)].
For incommensurate ratios of TD and TR the SSPM
become much more complex. Still we are able to recover
the modes which satisfy the PSC at the pulse arrival
times. In Fig. 2 (b) we show the SSPM at t = 0 and t =
TD for TD = 2TR/7 (K = 2 , L = 5 ), where the arrows
indicate the frequencies which satisfy the PSC for the
two pulse train. Only a small number of such modes fall
within the average distribution of electron spin precession
modes, because the distance between the PSC modes is
proportional to 2πK/TD = 2π(K +L)/TR . The diluted
spectra of PSC modes for incommensurability decrease
the magnitude of the FR bursts between the pump pulses,
in accord with experiment. This shows, that although
any ratio of TD/TR can be satisfied by large K and
L , the FR signal between the pulses should be negligibly
small in this case. Consequently, not any ratio of TD/TR
leads to pronounced FR bursts.
To obtain further insight into the tailoring of electron
spin coherence, which can be reached by a two-pulse
train, we have turned from co- to counter-circularly po-
larized pumps. The delay between pumps TD was fixed
at TR/6 ≈ 2.2 ns. The time dependencies of the corre-
sponding FR signals are similar, as shown in Fig. 3. Be-
sides the two FR bursts directly connected to the pump
pulses, one sees a burst +1 due to CI of spin synchro-
nized modes. The insets in Fig. 3 (a) show closeups of
the different FR bursts. The sign, κ , of the FR am-
plitude for the counter-circular configuration undergoes
2TD -periodic changes in time relative to the co-circular
case, as seen in Fig. 3 (b), which demonstrates optical
switching of the electron spin precession phase by π in
an ensemble of QDs.
The observed effect of sign reversal is well described
by our model. Let us consider first a two-pulse train
with delay time TD = TR/2 for which the two pumps
-1 0 1 2 3 4 5
0 1 2 3 4 5
 co, 
 counter, 
 Time (ns)
pump 1 pump 2 +1 burst
- (b)  
 Time (ns)
B = 3 T T = 6 K (c)
FIG. 3: (a): Faraday rotation traces in the co-circularly (blue
traces) or counter-circularly (red traces) polarized two pump
pulse experiments measured for TD = 2.2 ns and B = 6T.
Insets give close-ups showing the relative sign, κ , of the FR
amplitude between the two traces. κ is plotted in (b) vs
time. (c): Effect of temperature on the FR amplitude in two-
pump-pulse experiment. TD = 1.88 ns.
are counter-circularly polarized. In this case an electron
spin can be synchronized only if at the moment of pulse
arrival it has an orientation opposite to the orientation
at the previous pulse. This leads to the PSC ωe =
2π(N + 1/2)/TD . The contribution of such precession
modes to the electron spin polarization is proportional
to cos[2π(N+1/2)t/TD)] = cos(2πNt/TD) cos(πt/TD)−
sin(2πNt/TD) sin(πt/TD) . Summing these contribu-
tions, only the first term gives a CI, whose modulus has
period TD , while the sign of cos(πt/TD) changes with
period 2TD . Only each third of the precession frequen-
cies can be synchronized by a counter-circularly polarized
two pulse train when the delay time is TR/6 as in our ex-
periment. However, the corresponding PSC has the same
dependence on TD . The CI modulus also has period TD
and its sign changes with period 2TD . The relative sign
of the FR amplitude for the counter- and co-circularly
case, κ = sgn{cos[πt/TD]} , is in accord with the exper-
imental dependence in Fig. 3 (b).
The CIs of the electron spin contributions can be seen
only as long as the coherence of the electron spins is main-
tained. In this respect the temperature stability of the
CI is especially important. Fig. 3 (c) shows FR traces
in a two-pump-pulse configuration with TD = 1.88 ns at
different temperatures. For both positive and negative
delays, the FR amplitude at a fixed delay is about con-
stant for temperatures up to 25 K, irrespective of slight
variations which might arise from changes in the phase
synchronization of QD subsets. Above 30 K a sharp drop
occurs, which can be explained by thermally activated
destruction of the spin coherence.
The electron spin coherence in charged QDs is initi-
ated by generation of a superposition of an electron and
a charged exciton state by resonant pump pulses [17, 19].
The simultaneous decrease of the FR magnitude before
each pump pulse and afterwards (when the CI signal is
controlled by the excitation pulse) suggests that the co-
herence at elevated temperatures is lost already during its
generation. The 30K temperature threshold corresponds
to an activation energy of ∼ 2.5 meV. This energy may
be assigned only to the splitting between the two lowest
confined hole levels, because the electron level splitting
dominates the 20 meV splitting between p- and s-shell
emission in photoluminescence and is much larger than
2.5 meV. The decoherence of the hole spin results from
two phonon scattering, which is thermally activated and
should occur on a sub-picosecond time scale, i.e. within
the laser pulse [20]. The fast decoherence of the hole spin
at T > 30 K suppresses formation of the electron-trion
superposition state. ps-pulses as used here are therefore
not sufficiently short for initialization of the superposi-
tion and creation of a long-lived electron spin coherence.
In summary, we have demonstrated that the mode-
locking effect allows a far-reaching control of electron
spin coherence in QD ensembles during the spin coher-
ence time of microseconds [1]. Two-pulse train mode-
locking selects QD subsets which give a non-dephasing
contribution to the ensemble spin precession. The tech-
nique shows remarkable stability with respect to temper-
ature increase up to 25 K, a property which is important
for utilizing it in quantum information processing. The
robustness of this control technique is provided by the
dispersion of the spin precession frequencies in the QD
ensemble.
Acknowledgments. This work was supported by the
BMBF program nanoquit, the DARPA program QuIST,
the ONR, the DFG (FOR485) and FAPESP.
[ † ] on leave from the Instituto de Fisica Gleb Wataghin,
Campinas, SP , Brazil.
[ § ] also at Ioffe Physico-Technical Institute, 194021, St. Pe-
tersburg, Russia.
[ ‡ ] also at Institute of Physics, St. Petersburg State Univer-
sity, 198504, St. Petersburg, Russia.
[ ⋆ ] also at School of Computational Sciences, George Mason
University, Fairfax VA 22030.
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http://arxiv.org/abs/cond-mat/0610152
http://arxiv.org/abs/cond-mat/0610433
