Rapid and robust control of single quantum dots
Light: Science & Applications
Rapid and robust control of single quantum dots
Pablo M de Roque
Niek F van Hulst
The combination of single particle detection and ultrafast laser pulses is an instrumental method to track dynamics at the femtosecond time scale in single molecules, quantum dots and plasmonic nanoparticles. Optimal control of the extremely short-lived coherences of these individual systems has so far remained elusive, yet its successful implementation would enable arbitrary external manipulation of otherwise inaccessible nanoscale dynamics. In ensemble measurements, such control is often achieved by resorting to a closed-loop optimization strategy, where the spectral phase of a broadband laser field is iteratively optimized. This scheme needs long measurement times and strong signals to converge to the optimal solution. This requirement is in conflict with the nature of single emitters whose signals are weak and unstable. Here we demonstrate an effective closed-loop optimization strategy capable of addressing single quantum dots at room temperature, using as feedback observable the two-photon photoluminescence induced by a phase-controlled broadband femtosecond laser. Crucial to the optimization loop is the use of a deterministic and robust-against-noise search algorithm converging to the theoretically predicted solution in a reduced amount of steps, even when operating at the few-photon level. Full optimization of the single dot luminescence is obtained within ~ 100 trials, with a typical integration time of 100 ms per trial. These times are faster than the typical photobleaching times in single molecules at room temperature. Our results show the suitability of the novel approach to perform closed-loop optimizations on single molecules, thus extending the available experimental toolbox to the active control of nanoscale coherences. Light: Science & Applications (2017) 6, e16239; doi:10.1038/lsa.2016.239; published online 10 March 2017
closed-loop control; coherent control; single quantum dot; two-photon luminescence; ultrafast
Spectroscopy and microscopy at the single molecule level provide the
most detailed insight in complex environments, revealing
unprecedented detail about molecular structure and dynamics even at ambient
conditions1?4. Indeed, applications of single molecule spectroscopy are
numerous, ranging from the determination of molecular interactions
in biological systems5 to the development of super-resolution
microscopy techniques6, and the implementation of quantum optics
protocols in the solid state7. The next step is to facilitate single
molecule spectroscopy with ultrafast laser pulses and thus gain access
to the intrinsic ultrafast molecular processes such as vibrational
motions, excitation energy transfer, charge transfer and electronic
dephasing, all of which have femto- to picosecond characteristic times,
with the major advantage to overcome the loss of information
inherent to conventional ensemble pump-probe measurements4.
To capture the ultrafast dynamics in single molecules and
nanoparticles, one needs to excite with sophisticated well-tailored pulse
sequences, generally referred to as ?coherent control?. The field of
coherent control8?12 has developed two generic approaches in this
context, open-loop and closed-loop control. Both strategies are being
applied to ensembles of systems, yet the step towards the optimal
control of single nanoparticles has remained challenging and is the key
point of this paper.
The open-loop approach relies on optimal control theory, in which
the ?best? laser pulse sequence for inducing a desired response is
calculated a priori from the assumed system?s Hamiltonian9,13. For
complex systems like molecules and nano-emitters in dense
environment and at room temperature, complete a priori calculations are
often very complicated or even impossible. For this reason, in recent
works, the coherence properties of single molecules and energy
transfer complexes were investigated using simplified and more
intuitive electric fields?for example, a sequence of two pulses whose
relative phase and time delay could be varied?chosen based on the
ensemble absorption spectra14?16. Even if these simplified fields led to
some insight into the coherence of the molecules investigated, they
were not optimal for controlling the ultrafast dynamics, especially
given that for each single molecule the spectrum does deviate from the
The closed-loop approach, often the preferred one to control
arbitrarily complex systems, consists in resorting to an adaptive
closed-loop feedback optimization following the seminal paper by
Judson and Rabitz18. A specific experimental observable (for example,
one-photon- or multi-photon-induced fluorescence in molecules or
second harmonic (SH) generation in crystals) guides a learning
algorithm that varies the spectral phase and time profile of the laser
pulses. Closed-loop control schemes were successfully applied to
ensembles of different systems19?23, yet the ultimate target of
extending this approach to single molecules has remained challenging.
Closed-loop approaches typically need many iterations and strong
signals to converge to the optimal solutions4. However, these
requirements are generally in conflict with the nature of single molecules and
nano emitters. As one moves from ensemble measurements to the
investigation of single systems, the observable signal reduces to the few
photon level with a shot-noise that can be of the order of several
percent for ideal single quantum emitters24. To overcome this
difficulty, one needs to increase the integration time for every step
in the optimization process, which considerably extends the
experimental time. On the other hand, photo-induced processes?such as
blinking and photobleaching?that molecules and quantum dots
(QDs) undergo at room temperature25, together with experimental
drifts that become particularly important when measuring on single
emitters and nanoparticles, limit the total observation time to few
minutes at best.
A recent experiment demonstrated enough sensitivity to attempt a
closed-loop control experiment on single molecules in the linear
weak-excitation limit26. By using a genetic algorithm, similar to those
used in bulk measurements, no optimization could be obtained and
the fluorescence signal from the molecules remained unchanged.
Learning algorithms that work well for ensembles are, in general, not
applicable to the investigation of weak single photon emitters. For a
closed-loop algorithm to work for single molecules at room
temperature, two requirements need to be fulfilled: (i) the trial phases applied
by the algorithm need to produce an effect larger than the intrinsic
(shot) noise of the signal; (ii) the optimal solution has to be found in a
reduced amount of trials (time) to overcome experimental drift and
photobleaching of the molecules.
Here we demonstrate an improved all-optical closed-loop
phasecontrol scheme capable of rapidly optimizing the
photoluminescence of a single quantum emitter at room temperature, thus
satisfying both the above conditions. The method is demonstrated
on single QDs at room temperature using as feedback variable,
the two-photon-induced photoluminescence (TPPL). The spectral
phases from broadband femtosecond laser pulses are
deterministically varied based on a multiple one-dimensional search (MODS)27
that allows rapid global optimization of the observable using
reduced amount of trials even at the few photon level. This
approach is possible because in the case under study no local
minima exist that can hamper the optimization, as explained below.
Because the theoretical dependence of the two-photon absorption
(TPA) process on the spectral phase of the incident field in these
systems is well known12,28, the experiment allows us to validate the
correctness of the found solution.
MATERIALS AND METHODS
A schematic of the experimental procedure is shown in Figure 1.
In short, a 20-fs-laser pulse (central wavelength 800 nm, bandwidth
100 nm) entered a phase-only pulse shaper, arranged in a 4-f
configuration, in which a spatial light modulator (SLM) made up of
two liquid crystal masks of 640 pixels each (SLM-640 CRi) controlled
the spectral phase of the laser pulse. The laser beam was then sent
through an oil-immersion microscope objective (1.4 numerical
aperture) and focused on the sample plane, inducing TPA in the
studied QDs. We used colloidal CdSe/CdS core/shell QDs
characterized by a 4.8 ? 15 nm CdSe core rod embedded in a CdS
shell, yielding an overall QD size of 9.8 nm by 44 nm. They were
synthesized according to the procedure described in Ref. 29, dispersed
in toluene and finally spin-coated in a PMMA matrix on a
180-?m-thick glass coverslip for microscope imaging. Due to the
large volume of the CdS shell, these QDs are known to be efficient
two-photon absorbers30?32. In these systems, the total volume
determines the cross-section for TPA and is given mostly by the CdS shell
volume, whereas the exciton confinement and thus the emission
spectrum can be tuned independently by changing the dimensions of
the CdSe core. Our QDs have an absorption spectrum that is
dominated by the CdS shell absorption with an onset at 515 nm,
and a Stokes-shifted luminescence emission centered at 670 nm.
The resulting TPPL emitted by the QDs was collected through the
same objective, detected with an avalanche photodiode in Geiger
mode after spectrally filtering the signal and constituted the feedback
variable for the MODS algorithm. The algorithm changed the spectral
phase on the SLM and searched for the phase mask that maximized
the TPPL signal. In this manuscript, we show that it is experimentally
feasible to perform such closed-loop TPPL maximization even on
RESULTS AND DISCUSSION
We present the main result of this work in Figure 2, namely the
realization of a phase-only closed-loop optimization of the TPPL
emitted by single QDs at room temperature. First, we acquired a
confocal image by raster scanning the QD sample while recording the
TPPL signal (Figure 2a). Due to the low concentration of used QDs,
we could spatially resolve isolated bright spots corresponding to single
QDs or small clusters of QDs. We further chose QDs presenting low
TPPL signal (circled QDs) as candidates for the optimization
experiment on single emitters. Photon correlation measurements performed
on these QDs confirm that these were non-classical emitters, as can be
Figure 1 Schematic of the experiment. Laser pulses from a broadband
femtosecond laser induce two-photon absorption in rod-shaped CdSe/CdS
quantum dots, after passing through a phase-only pulse shaper. TPPL signal
collected using an oil-immersion objective is spectrally filtered and sent to a
single photon avalanche photodiode (APD). For each optimization step, the
multiple one-dimensional search optimization algorithm searches for the
spectral phase on the spatial light modulator pixels that maximizes the TPPL
signal using it as feedback variable.
?300 ?200 ?100 0 100
Time delay (ns) QD1 QD2 QD3
deduced from the photon anti-bunching dip at time zero in the
graph of Figure 2b. For these measurements, we used a helium?
neon laser that linearly excited the QDs; we subsequently divided
the collected photoluminescence from the QDs with a beam
splitter and sent it to two different avalanche photodiodes arranged
in a Hanbury-Brown and Twiss intensity interferometer
configuration. We note that the second-order photon autocorrelation
traces were neither corrected for accidental coincidences arising
from detector dark counts nor for the presence of biexcitons in
QDs. Thick-shell QDs, like the ones we used, present high
biexciton photoluminescence quantum yields33. This has the effect of
increasing the normalized coincidences at zero-time delay up to
half for high-excitation powers as the ones used in these
experiments34. After selecting the suitable QDs behaving as single
emitters, the MODS optimization algorithm was used to optimize
the TPPL signal on the chosen QDs. We present optimization
routines in Figure 2c, showing the TPPL for three different QDs
for every step of the optimization algorithm (cf. algorithm
description). As different spectral phases were applied by the
SLM in a deterministic way, the TPPL signal increased until it
reached a maximum value. For all the investigated QDs, the TPPL
increased by a factor between 4 and 6.
under the following
Low-signal conditions. For these QDs, the typical TPPL intensity
detected by the avalanche photodiode was in the range 2000?7500
photons per second with an average excitation power of 20?50 ?W,
which is comparable to the power and signal levels for typical single
Few total acquisitions. With a total number of only 300 acquisitions,
the algorithm was able to find the best solution. In order to have
sufficient signal-to-noise ratio, we used typical integration times of
100?800 ms for each acquisition. This means that the full
maximization was carried out in about a minute, which is in stark
contrast to orthodox optimization strategies based on evolutionary
algorithms that typically perform hundreds of generation steps with
a pool of hundreds of measurements per generation. Hence, our
algorithm typically needs two orders of magnitude fewer
measurements to find an optimum.
On the basis of these numbers, closed-loop coherent control
experiments even in single molecules at room temperatures become
possible. Our optimization scheme is sufficiently robust to handle
signals on the few photon level and at the same time sufficiently rapid
to beat photobleaching and experimental instability, which take place
on a few minutes timescale.
Since the maximization algorithm is crucial to the feasibility of the
experiment, we now describe in detail the strength of this strategy. In
general, the objective of closed-loop coherent control experiments is to
find the optimal set of control variables that maximize the signal of a
particular observable of the system under study. In our case, the
control variable is the spectral phase of a broadband femtosecond laser
pulse, and the observable is the TPPL signal of individual QDs. We
remark that we are dealing with QDs that emit a limited number of
photons. The intrinsic noise of these systems, roughly the square root
of the number of photons, is of the order of 1% of the signal, in
addition to the stability of the optical setup (laser and microscope)
that, especially when looking at single emitters, can produce signal
drifts over time also of the order of a few percent. Hence, the change
produced by different sets of control variables on the observables need
to be larger than the intrinsic noise in the measurement.
Our femtosecond-laser pulse shaper incorporates a SLM with 640
pixels (control dimension), with 4096 different phase-delay values at
each pixel. It is obvious that performing a brute-force global search
over such a vast solution landscape is unrealistic due to the long time
required to perform such an experiment and the constraints of our
particular QD sample.
The MODS algorithm, whose working principles are illustrated in
Figure 3 (and explained in detail in the figure caption), is based on two
0 ? 2p
100 150 200
general assumptions. (
) The time profile of the laser field can be
expressed as a linear superposition of its frequency components, which
stems from the property of the Fourier transform; (
) in the majority
of the quantum control landscapes there are no local traps, namely no
local maxima that differ from the global maximum8,35,36. The first
assumption allows separating the maximization problem in simpler
one-dimensional maximization problems, one for every frequency
component. In practice, since different SLM pixels (16 in this case) are
grouped together in the experiment to form 40 control nodes, the
search for the optimal phase mask is reduced to 40 one-dimensional
searches. For each node, the algorithm searches for the phase value in
the 0?2? domain (4 different tries are performed in the experiment
described) that maximizes the targeted observable (TPPL from single
QDs), while keeping the rest of the nodes fixed. Once the optimal
phase value for a node has been found, the algorithm fixes its value in
the mask and repeats the same procedure in the next node. Because in
the case under study, as in many different quantum control
problems35, no local traps exist, there is only one real maximum for
every node once the previous are set. This is the reason why the
optimization progresses in a very efficient way towards the global
maximum, even when acting on low-signal levels.
The number of nodes used, as well as the number of different
phases applied at each node by the algorithm, which are called phase
jumps, is very critical for the success of the optimization and should be
carefully chosen as a compromise between two counteracting effects.
On one hand, the number of nodes determines the sampling rate of
the algorithm or the effective optical resolution at the SLM. A bad
optical resolution (low number of nodes used) in turn implies two
great limitations in the phase functions that the algorithm can
produce11. First, phase functions varying more rapidly than the
sampling rate cannot be generated, or, in the time domain, delays
longer than the inverse of the optical resolution are beyond the shaper
capability. Thus, the number of nodes determines the magnitude of
the search space that the algorithm is restricted to. However, for most
solid-state quantum systems at room temperature, the coherence time
is limited to few tens of femtoseconds at best. Long time delays are
therefore, in general, not needed when performing coherent control in
these cases. Second, when using a limited amount of nodes, smooth
phase functions can only be approximately reproduced. In this case, a
bad optical resolution generates jagged phase functions and the effect
is amplified when low numbers of phase jumps are used. This
limitation implies that the best solution might only be approximated
to a certain level of precision. In order to maximally extend the search
space and produce smooth phase functions, the number of nodes and
phase jumps has to be as high as possible. On the other hand, the
frequency band of a node, and the amount of phase variation between
consecutive phase jumps, also determine how large the total change in
the feedback variable (in this case the TPPL from a QD) will be when
varying the phase of that node in the search for the maximum. As
discussed above, an important criterion for the optimization success is
that the variation of the feedback signal produced by the application of
different phases should be larger than the intrinsic noise of the system.
Too large amount of nodes, that is, a too small frequency band
associated to every node, or too many phase jumps would result in
only small variations of the feedback signal, which might hamper the
optimization. As shown below (see also Supplementary Information),
in the case of the current experiment, a good compromise between the
two effects could be found. By using 40 nodes and four phase jumps,
we ensured at the same time that the TPPL signal could be quickly
optimized, and that the MODS algorithm could find the best solution
to a good approximation (with 80% precision, as demonstrated
In order to prove the correctness of the optimization experiment
presented here, we need to demonstrate that the solution found by the
MODS algorithm is indeed close enough to the real optimum for
the QDs studied. The algorithm starts with an initial phase mask on
the SLM and produces a final phase mask that optimizes the TPA.
Looking at the difference between initial and final phase masks, we can
compare the results of this experiment with theoretical predictions.
The phase difference, in the case of the experiments illustrated in
Figure 2c, is plotted as a blue curve (called the Algorithm Phase) in
Figure 4a and has a well-defined shape.
To understand this phase, we separately measured the initial
spectral phase of the laser pulse with the multiphoton intrapulse
interference phase scan (MIIPS)37 on a SH crystal, plotted as a red
curve in Figure 4a. This curve represents the spectral phase of the laser
pulse when the starting phase mask on the SLM was applied.
As discussed in literature37, a non-flat spectral phase, as the red curve
of Figure 4a, corresponds to a distorted pulse, which is not the shortest
possible for a given laser spectrum, that is, it is not a transform-limited
(TL) pulse. The similarity between the red and the blue curves implies
that the optimization process can be thought of as an attempt to
produce TL pulses at the position of the QDs. This is confirmed by
pulse autocorrelation measurements corresponding to nearly TL
400 500 600 700
Wavelength (nm) TPA
TPPL 100 x Laser 800
pulses (Supplementary Information). The same experiment was
repeated with different starting phase masks and the maximization
process always ended up close to the TL optimization point (see
In order to understand the outcome of the experiment, in
Figure 4b, we report the absorption spectrum of the QDs studied,
which is dominated by the CdS, having an onset around 515 nm and
increasing for smaller wavelengths. As the graph shows, there is no
overlap between the laser spectrum and the tail of the QDs absorption,
meaning that the fundamental laser cannot induce any single photon
transition in the QDs and only pure two-photon transitions around
400 nm (in the region of the SH spectrum of the laser) are possible.
Moreover, at the SH wavelength, the absorption spectrum of the QDs
is very broad, and no discrete state is expected to be resolved at room
temperature even in a single QD. In other words, the QDs absorb all
the wavelengths within the SH spectrum of the laser with a similar
efficiency. As can be theoretically demonstrated11,38, in these
conditions, namely for very broad absorbers at the SH wavelength with no
real intermediate state, the TL pulse is the unique solution that
maximizes the TPA. By choosing these QDs, we therefore can verify
that the algorithm converges to the theoretically predicted solution,
which demonstrates the correctness of our closed-loop experiment.
As shown in Figure 4c, this optimization experiment may be seen as
the simplest example of coherent control, based on the interference of
the different possible sum frequency processes (pathways) leading to
the same final state8,12. Non-resonant two-photon transitions for a
broadband field involve many routes through a continuum of virtual
levels, where the interference pattern excited by the
multiplefrequency components can enhance or diminish the total transition
probability. Since the interference effect depends on the spectral phase
distribution of the laser field, controlling the spectral phase gives
access to the control of the two-photon transition probability8.
We want to address now the limitations of the current approach,
both in terms of the search space and the precision of the found
solution. The applicability of this scheme to real coherent control
experiments on the nanoscale depends on how severe these
As discussed above, the magnitude of the search space is limited by
the optical resolution. In this case, for a laser bandwidth of ~ 100 nm,
centered around 800 nm, dividing the spectrum on 40 nodes
corresponds to an effective time window for the algorithm to operate
of ~ 200 fs. The initial pulse, namely the pulse that corresponds to the
MIIPS phase of Figure 4a, is about 100 fs long (see simulations in
Supplementary Information), therefore well within the search space of
the algorithm, which is why the optimization can converge to a
solution close to the real optimum (the TL pulse). In many cases of
interest, for instance for molecules14, light-harvesting complexes16,
QDs39 and plasmonic nanoparticles40,41 at room temperature, the
coherence time was measured to be shorter than 100 fs. The current
implementation of the MODS algorithm would therefore lead to a
large enough search space to make coherent control of single
molecules and nanoparticles at room temperature possible, in all the
problems for which conditions (
) and (
) for the applicability of the
approach are satisfied.
As shown in Figure 4a, the Algorithm Phase does not perfectly
coincide with the MIIPS phase. In order to address the precision of the
found solution, we also measured the increase in the SH signal
obtained after the MODS optimization and compared it with what the
MIIPS algorithm obtains (see Supplementary Information). The
increase in the SH signal obtained with the current implementation
of the MODS algorithm was always about the 80% of what the MIIPS
method could obtain. In other words, the best solution could be found
very rapidly, with an 80% precision when maximizing on a single QD.
Considering that MIIPS is a dedicated method for compressing pulses
and takes advantage of the good theoretical knowledge of nonlinear
optical processes, whereas the MODS algorithm does not need any
preknowledge and can optimize on arbitrary signals, we believe that
this is a very remarkable demonstration of closed loop optimization on
single-quantum emitters that almost reaches the best solution.
Furthermore, the MODS algorithm does not need any spectral
information to work, as opposed to the MIIPS algorithm that needs
the complete SH spectrum. Refining the solution with successive
iterations, using higher number of nodes and phase jumps would lead
to a better optimum, but would also require longer measurement
times, which might become incompatible with the nature of single
With the current experiment, we pushed closed-loop optimization
techniques to the limit of single-quantum emitter sensitivity,
demonstrating maximization of the TPPL emitted by single QDs
at room temperature. At the same time, our strategy satisfied all
the conditions required to achieve closed-loop coherent control of
single molecules at room temperature: the signal levels were as low
as the ones found in single molecule experiments; the TPPL
optimization was performed in o4 min (faster than typical
photobleaching times), made possible by a speed up of about
two orders of magnitude over widely used traditional evolutionary
strategies; the algorithm succeeded in finding a solution that well
reproduced the theoretical expectation.
The fact that, in our experiment, the best solution for the
targeted problem is the TL pulse, used here to validate our
approach, is a consequence of the physics of the QDs studied
and does not limit the general applicability of the method. Applied
to different problems with different underlying physics, our
optimization strategy would maximize the targeted observable by
inducing constructive interference between the pathways
connecting to the desired final state. Optimizations in systems,
characterized by non-trivial responses within the laser bandwidth would in
general lead to optimal solutions that differ from the TL pulse. We
speculate that our method can be applied to the study and control
of plasmonic fields in complex nanostructures42; the steering of
coherences in individual molecular systems with specific laser
fields16,26, allowing time-resolved quantum state tomography of
molecular systems43; and the selective excitation of different
absorptive species with high spatial resolution39?44.
CONFLICT OF INTEREST
The authors declare no conflict of interest.
This research was funded by the European Commission (ERC Adv. Grant
247330-NanoAntennas and ERC Adv. Grant 670949-LightNet), Spanish Severo
Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0522), Plan
Nacional Project FIS2012-35527, co-funded by FEDER, the Catalan AGAUR
(2014 SGR01540) and Fundaci? CELLEX (Barcelona). PMdR acknowledges
financial support from Spanish Government MINECO-FPI grant and European
Science Foundation under the PLASMON-BIONANOSENSE Exchange Grant
program. MGS acknowledges financial support from grants MICINN
TEC2011-22422 and MINECO TEC2014-52642-C2-1-R.
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Supplementary Information for this article can be found on the Light: Science & Applications? website (http://www.nature.com/lsa).
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