# Luminescence properties of closely packed organic color centers grafted on a
carbon nanotube

Antoine Borel  Federico Rapisarda  Laboratoire de physique de l’ENS,
Université PSL, CNRS, Sorbonne Université, Université Paris Cité,75005 Paris,
France  Stephen K. Doorn  Center for Integrated Nanotechnologies, Materials
Physics and Applications Division, Los Alamos National Laboratory, Los Alamos,
New Mexico 87545, United States  Christophe Voisin  Yannick Chassagneux
<EMAIL_ADDRESS>Laboratoire de physique de l’ENS, Université
PSL, CNRS, Sorbonne Université, Université Paris Cité,75005 Paris, France

###### Abstract

We report on the photo-luminescence of pairs of organic color centers in
single-wall carbon nanotubes grafted with 3,5 dichlorobenzene. Using various
techniques such as intensity correlations, super-localization microscopy or
luminescence excitation spectroscopy, we distinguish two pairs of color
centers grafted on the same nanotube; the distance between the pairs is on the
order of several hundreds of nanometers. In contrast, by studying the strong
temporal correlations in the spectral diffusion in the framework of photo-
induced Stark effect, we can estimate the distance within each pair to be of
the order of a few nanometers. Finally, the electronic population dynamics is
investigated using time-resolved luminescence and saturation measurements,
showing a biexponential decay with a fast overall recombination (compatible
with a fast population transfer between the color centers within a pair) and a
weak delayed repopulation of the traps possibly due to the diffusion of
excitons along the tube axis.

###### keywords:

Carbon nanotube, Organic Color Center, Spectral Diffusion, Coupled Emitters

##  1 Introduction

Color centers stand for pseudo atomic systems embedded in a solid-state
material that can be manipulated optically for coherent control, single photon
emission and other quantum operations 1, 2, 3. They stand up as valuable
building blocks for quantum technologies, in particular for all the
technological advantages associated to solid-state systems. Among many
possible approaches for quantum manipulations, the case of quantum emitters in
close vicinity ($\delta\sim\text{nm}$) is of particular interest for sensing
4, 5 or to implement quantum superpositions and entangled states yielding for
instance super and sub-radiance effects… Such effects have been demonstrated
using incidental pairs of quantum dots 6 or pairs of molecules embedded in a
matrix at low temperature 7, 8. In this respect, the control of the relative
position of the emitters at the nanoscale, at least along one or two
directions, would open the way to a better control over the fabrication of
coupled quantum emitters, for applications in solid-state quantum
technologies.

Carbon nanotubes (CNTs) have recently been considered for such applications
due to the possibility to use them as a functionalizable backbone to graft
various molecules by means of organic chemistry tools such as diazonium
chemistry 9, 10, 11, 12, 13. The molecule grafted on a semi-conducting carbon
nanotube acts as a deep potential well ($\sim 100$ meV) for the excitons
generated across the bandgap of the nanotube, yielding the concept of organic
color centers (OCC) which proved to be an original single photon source in the
near infrared (telecom bands O and C) up to room temperature 10. This 1D
backbone increases drastically the probability to generate pairs of closely
packed emitters (much closer than the wavelength $\lambda$) since it basically
scales like $\delta/\lambda$ using the 1D template whereas it scales like
$(\delta/\lambda)^{3}$ in a random 3D case, yielding a $10^{4}$ fold
enhancement. This concept was recently explored for guanine decorated CNTs,
where closely spaced OCCs could be observed, giving rise to anti-correlation
signatures 14.

In this work, we report on the observation of two couples of OCC along the
same nanotube, showing strong spectral correlations, which we analyse in the
framework of Stark induced spectral diffusion. We obtain an estimate of the
distance between OCCs within each pair of the order of a few nanometers. Using
saturation and time-resolved spectroscopy, we investigate the population
dynamics within each pair of OCC.

##  2 Methods

The sample consists in (7,5) single-wall carbon nanotubes (CNTs) dispersed and
wrapped in a polymer (PFO) in suspension in a 5 % mass polystyrene in toluene
solution. This solution is spin-coated on a Bragg mirror resulting in
individual (7,5) CNTs embedded in a 160 nm thick polystyrene layer. The CNTs
are then functionalized by 3,5 dichlorobenzene diazonium salt by the dip
doping method 10. The 3,5 dichlorobenzene ion is covalently attached to the
CNT surface by converting a $sp^{2}$ into a $sp^{3}$ bound. In this study, we
focused on the subclass $E_{11}^{*-}$ of the photoluminescent states of the
OCC, that emit in the O band 10.

The experimental setup consists in a home-made microscope with the objective
inside a closed-cycle cryostat. The sample is cooled at 9K (see details in 15,
16). The excitation laser at a wavelength of 800 nm is either a cw or pulsed
Ti:Saphire laser (tunable cw MSquared laser or pulsed Coherent Vitesse Laser).
The photo-luminescence (PL) was analyzed either with a spectrometer (IsoPlane
300) coupled to an InGaAs CCD (Princeton NIRvana) or with a fibered
superconducting single photon detector (Scontel SSPD). The CNT is coupled to a
fiber Fabry-Perot cavity, as described in 16. Here, the cavity is used
primarily as a sharp tunable band-pass filter.

Observations. We investigated a 20 meV wide spectral band around 0.93 eV, in
the telecom O band. Beside numerous spectra consisting in a single line, we
happened to observe more complex spectra such as that of Fig. 1(b), which
consists of 4 lines. The evolution of the spectrum over time (with a 1 s time
bin) is shown in Fig.1 (c). Correlated spectral wandering and jumps can be
observed. Quantitative assessment of the correlations between these lines can
be gained by computing the correlation matrix of the spectral position of the
four lines over time Fig.1 (d). The emission energies are temporally
correlated for lines $A_{1}$ and $A_{2}$ (Fig.1 (e)) in the one hand and lines
$B_{1}$ and $B_{2}$ on the other hand (Fig.1 (f)).

Such correlations may have several origins. They are either the signature of
two very close emitters sharing the same electrostatic environment and thus
having correlated Stark shifts, or the signature of a single emitter showing
two bright levels or a single emitter subject to fast switching between two
energy levels (much faster than the detection time scale, hence showing up as
two lines) due to the interaction of a single emitter with a nearby fast
fluctuator. In the following, we show that the spectrum of Fig. 1(b) actually
arises from two pairs of closely packed emitters as depicted in Fig. 1(a).

Eventhough the dark/bright splitting reported for both pristine and aryl-
functionalized (7,5) SWCNT (4 meV to 15 meV) under high magnetic field ($>$ 5
T) 17, 18 roughly matches the splittings we observe for pair $A$ and pair $B$
(Fig. 1(b)), we rule out the interpretation because in the absence of a
magnetic field the intensity contrast would be much larger than observed here.
In addition, the observation of such split lines in our sample is very
unusual, at contrast with an intrinsic dark/bright splitting which should be
ubiquitous. A fluctuator - such as a charge or a dipole switching rapidly
between two states or positions in the close environment of the OCC - could
split the excitonic emission energy on a time scale much shorter than the
acquisition time. The fluctuations of these lines on the longer (1 s) time-
scale would naturally be correlated since both lines arise from the same
microscopic object. In addition, the intensity auto-correlation function
should show a signature at the timescale of this switching. More specifically,
the autocorrelations of both lines should exhibit a step at the same time
delay (an exponential decay appears as a step when the time axis is
represented in logarithmic scale); the product of the two step amplitudes
being equal to one (see model in SI). Fig. 2(a) shows the auto-correlations of
each of the four lines, over a time span between 60 ps and 1 s (semi-log
scale). Although steps can be identified, the amplitudes in B1 and B2 or in A1
and A2 do not fulfill the condition described above, ruling out the hypothesis
of a fast fluctuator. Note that the presence of oscillations in the
autocorrelations for times exceeding 0.1 ms are signatures of mechanical
vibrations in our setup due to the high sensitivity of the Fabry-Perot cavity
used for spectral filtering. In addition, for such fast switching scenario,
one would also observe cross-correlations in the intensity fluctuations or
cross-correlations between the intensity and the energy position corresponding
to the slow variations of the charge state of the emitter 19. This is not the
case here, as showed by the correlation matrices (SI). Finally, we show below
that each line has its own decay dynamics, which tend to confirm that they
actually arise from physically different OCCs. Hence, we rule out the scenario
of a fast switching system and we are left with the simplest explanation :
each pair of lines arises from two distinct OCCs that are close enough to
probe the same fluctuating electrostatic environment.

Figure 1:  (a) Sketch of the system (b) Typical luminescence spectrum
(excitation wavelength 800 nm, power density 4.2 kW/cm2 showing four lines
within a 20 meV window. (c) Time trace of the PL spectrum with a 1 s time bin
showing correlated spectral wandering and spectral jumps. (d) correlation
matrix of the fluctuating emission energies of the four lines as labeled in
(b) computed from temporal traces (c). (e) and (f) time trace of the energy
shifts for the emitters in pair A and pair B respectively.

The diffraction limited resolution of the microscope is not sufficient to
decide whether the four OCC actually belong to the same nanotube or to
distinct ones (within the diffraction spot). To investigate this question
further, we first checked their polarization diagrams. In fact, due to the
strong aspect ratio of the nanotube, depolarization effects strongly polarize
both their intrinsic PL and that arising from nearby emitters 20 along the
nanotube axis. The diagrams are shown in Fig. 2 (c-f), open circles, together
with a dipolar diagram fit. The four spectral lines are co-polarized in
emission with the same angle $14^{\circ}\pm 4^{\circ}$ (with respect to an
arbitrary horizontal direction). Interestingly, the same copolarization
signature is observed on the excitation diagrams (full symbols in Fig. 2(c-f),
meaning that the exciton generation is favored for the same polarization
orientation for all the lines. This strongly suggests that the four OCCs are
attached to the same nanotube and fed by the excitons generated at higher
energy in the nanotube. To confirm this interpretation, we performed an
excitation resolved PL study (PLE). The excitation spectrum of each line in
displayed in Fig. 2(g). The four emitters show the same resonances especially
at about 1.56 eV, in line with a common feeding process from the intrinsic
excitons of the nanotube. Finally, we performed a spectrally resolved super-
localization measurement in the spirit of the work by Raynaud et al. 21 in
order to possibly determine a sub-wavelength localization of the emission
sites. The spatial precision of such measurement relies on a trade-off between
the integration time and the signal-to-noise ratio and was typically 15 nm to
25 nm for optimized setups such as in 21 or 22 . Our setup was not design for
such experiment and we could only resolve the brightest emitters $A_{1}$ and
$B_{2}$ and we measured a distance of 300 nm $\pm$ 150 nm between them (see
SI). The distance within the pairs could not be resolved with this technique.

##  3 Results and discussion

Correlated Stark shifts. We used the correlated Stark shifts to evaluate the
typical distance between the two emitters. Within the framework of the linear
Stark effect, the energy shifts are given by: $\Delta E=-\vec{p}\cdot\vec{E}$,
where $\vec{E}$ is the electric field and $\vec{p}$ a permanent dipole
associated with the OCC. Glückert et al. 23 showed that an oxygen OCC has a
permanent dipolar moment of the order of 0.15 e.Å (where $e$ is the elementary
charge). This permanent dipole (carried by the OCC) has to be differentiated
from the transition dipole. In particular, it is not necessarily oriented
along the CNT axis but would rather point along the direction of the molecular
substituent making up the OCC. As a first approach, we thus computed the Stark
shifts using the permanent dipole moment of 3,5 dichlorobenzene which is of
0.47 e.Å (2 Debye).

Figure 2:  (a) Auto-correlation function under cw excitation of each lines
according to the color code of panel (b). (c-f) Polarization diagrams of each
PL line (open symbols for the PL signal full symbols for the excitation)
showing the copolarization of the four emitters. (g) Excitation spectrum of
each of the 4 lines according to the color code in (b) showing common
resonances at 1.56 eV and 1.265 eV.

This dipole is in interaction with the electrostatic environment. Using the
superposition principle, the trapping/detrapping of an elementary charge in
the vicinity of the dipole yields a Stark shift :

$\delta
E_{stark}=\pm\frac{e}{4\pi\epsilon_{0}\epsilon_{r}}\frac{\vec{p}\cdot\vec{r}_{p,e}}{r^{3}_{p,e}},$
(1)

where $\epsilon_{r}$ is the polystyrene static dielectric permittivity,
$\vec{p}$ is the permanent dipole moment, $\vec{r}_{p,e}$ is the vector
between the positions of the dipole and the charge, the positive (resp.
negative) sign corresponds to a trapping (resp. detrapping) event.
Experimentally, $\delta E$ represents the spectral shift between two
consecutive acquisitions. Indeed, we observe sharp spectral jumps which are
compatible with individual charge trapping events. In contrast, if many events
occurred within the integration time, one would rather observe smooth spectral
drifts.

In addition, we observed that the magnitude of the spectral diffusion
increases with the optical power, pointing towards photo-generated charges.
Since the polystyrene matrix is transparent at this wavelength, absorption
mainly occurs in the carbon nanotube, with charges being possibly transferred
to the close vicinity of the nanotube due to photo-ionisation. Hence, in a
first approach, we chose a quasi 1D model where the fluctuating charges are
bound to the nanotube axis. Hence, for each line, we computed the energy
difference between consecutive spectra. These energy differences are
represented in a bi-dimensional plot (Fig. 3(a,c)). To account for the 33 µeV
resolution of our spectrometer, we represented each point as a 2D Gaussian
spot with a standard deviation of 33 µeV. After summing up all these
contributions, we obtain a density plot representing the dual probability of
spectral jumps of the two lines of the pair (Fig. 3(b,d)). The two pairs show
contrasted distributions. For the pair A, the energy shifts $\delta E_{A1}$
and $\delta E_{A2}$ are linearly correlated with no observable deviation for
$|\delta E_{A2}|<0.5$ meV. For pair B, the density plot shows five main lobes
with symmetry center in (0,0). The central lobe also shows linear correlations
(a fit leads to a slope of 0.22 with only 0.01 errorbar). The pair of lobes
that are symmetrical with respect to the origin correspond to finite spectral
shifts of either identical signs (lobes Z2) or opposite signs (lobes Z3). We
modeled this distribution assuming that each point corresponds to the trapping
or de-trapping of an elementary charge in the vicinity of the nanotube
resulting in a shift given by eq.1. Small energy shifts are related to remote
charges and the slope of the central lobe can be related to the values of the
permanents dipoles or more precisely their projections along the nanotube
axis.

Figure 3:  (a) Spectral shifts for the emitters of pair A : each point
corresponds to the energy shift between consecutive spectra of line $A_{2}$
function of the shift of line $A_{1}$. 4672 points have been recorded. (b)
Density plot generated from panel (a) (see text). The blue, green and black
lines correspond to theoretical curves with dipole separation
$\delta=0,1\text{ and }2$ nm. (c) Similar to (a) for the emitters of pair B.
Five distinct clouds of points (symmetrical with respect to the origin) are
distinguishable hereafter called lobes Z1, Z2 and Z3. (d) similar to (b) using
the points of (c) as an input. The cyan and magenta lines correspond to dipole
separation $\delta=1\text{ and }4$ nm clearly indicating a finite separation
of the emitters.

For pair A, this analysis imposes an upper bound on the distance between the
two dipoles $\delta\lesssim 2$ nm. In fact, for larger dipole separation a
splitting of the central lobe is expected (dashed black line), due to an
increasing differential effect of a charge along the tube axis on the left and
right side OCC. In addition, the absence of side lobes (such as those of pair
B) also points to a low distance between the OCCs. In particular, the absence
of the anti-symmetric lobes rules out a configuration with a charge trapped
in-between the OCCs and thus points to a tiny separation within pair A.

For pair B, we can formally exclude that the two dipoles are located at the
same position since correlated shifts (lobes Z2) and anti-correlated shifts
(lobes Z3) are observed, which corresponds to the case where the trapped
charge is located between the two OCC. As explained above, the lobe Z1 results
from fluctuating charges located far away from the pair. Lobe Z2 corresponds
to a peculiar nearby trapping site on the CNT axis. We can deduce the dipole-
dipole separation in pair B to be of the order of 4 nm. In a 1D model, lobe Z3
would correspond to a single trapping site between the two dipoles. However,
in this configuration, the distances between the charge and the dipoles is
comparable to the diameter of the nanotube and the 1D approximation has to be
reconsidered. By considering the 3D geometry of the nanotube and different
dipole orientations, the spectral shifts observed in Fig.3(c,d) can be
reproduced. Examples of possible 3D configuration of the dipole on a nanotube
are presented in the SI.

We finally end up with the picture of two pairs of OCCs separated by a
distance of the order of 300 nm whereas the distance within each pair is on
the order of a few nanometers. This tiny separation between the OCCs in each
pair is likely to impact the dynamics of the system 14, 24, which we
investigate in the following section.

Population dynamics. To measure the PL decay, we exploit the coupling of the
nanotube to a tunable fiber Fabry-Perot cavity, as described in 16, allowing
us to obtain a good spectral rejection with a typical width of 50 µeV. After
filtering the excitation, the emission is sent to an SSPD detector. The PL
transients are shown in Fig. 4(c-f) together with bi-exponential fits (mono-
exponential fit for panel (f)). For both pairs of OCCs, the upper energy line
shows the fastest weighted average decay. For the pair $A$, the decay times of
each emitter are similar, yet the weight of the longer decay is 6 times
greater for the lower energy level. Note that the short decay time is below
our time-resolution. According to literature the lifetime of (7,5) aryl-
functionalized SWCNT is typically between 100-200 ps with a dominating
contribution of the longer component $w_{L}\simeq 0.9$ 10, 25 ($w_{L}$ being
the integrated weight of the long component). Emitters $A_{1}$ and $B_{1}$
(upper levels in each pair) thus exhibit unusually fast decay revealing
possible interaction with other levels. However, even line $A_{2}$ is
significantly faster than previously reported, especially regarding the
significant weight of the short decay component. In a first approximation,
$A_{1},A_{2}\text{ and }B_{1}$ show similar behavior with a dominant fast
decay (below our resolution of 10 ps) and a small weight long component with a
decay time of the order of 100 ps. For each pair of OCC, we model this
behavior by a four level scheme with a ground state, an upper reservoir level
pumped by the laser that populates the two OCCs levels corresponding to the
two lines of the pair (Fig. 4 (a)). Each level has its own decay time and we
include a possible population transfer between the levels. In a phonon
assisted process, this population mixing process is most likely strongly
asymmetrical due to the large energy splitting (compared to $k_{B}T$) within
each pair ($\sim 10$ meV). Hence this coupling simply results in reducing the
lifetime of the upper level, in agreement with the observations. However, this
coupling cannot explain the fast component of the lower level as it rather
results in a finite rise time, which we cannot observe due to our limited
time-resolution. The high energy excitation scheme used in our experiments is
assumed to populate both the OCC levels and a population reservoir
repopulating the OCC levels over a 100 ps time scale . This reservoir can be
either the $E_{11}$ excitonic level of the nanotubes (in agreement with
$E_{11}$ lifetime in the literature 15) due to exciton diffusion along the
tube axis, or from OCC dark states 25 , giving rise to the long decay observed
in all transient. This repopulation process does not contradict the high
purity of single photon emission of OCCs in the literature 10. In fact, the
weight of this mechanism is small and common OCCs have a longer lifetime than
this repopulation process, thereby substantially decreasing the likelihood of
double photon emission. The atypical value measured for the short decay
component, which thus corresponds to the OCC lifetime in our model, may be
caused by an unusually fast non-radiative decay for this specific OCC. In
contrast, $B_{2}$ can be accurately modeled by a mono-exponential decay of the
order of 150 ps. Here, it is important to note that if the OCC lifetime is
longer than the repopulation time from the reservoir, the overall decay is not
distinguishable from a mono-exponential decay, as is the case here, due to
limited signal-to-noise ratio. This also explains why $g^{(2)}(0)$ can be very
low for long-lived OCCs.

Figure 4: (a) Sketch of a pair of OCCs showing possible tunnel coupling. (b)
PL spectrum showing the color code for labelling the lines. (c-f) Semi-log
plot of the PL decay dynamics for the spectrally filtered lines, according to
the color code of (b) The black line corresponds to a bi-exponetial fits
(mono-exponential for (f) ). The lifetimes $\tau_{L}$ and $\tau_{S}$ were
extracted from the fits and $\text{w}_{\text{L}}$ indicates the integrated
weight of the long component. The instrument response function is indicated by
the grey line. (g) (resp. (h)) Saturation measurements of each line of pair A
(resp. B). For pair A (resp. B), the green (resp. orange) dashed lines of
panel (c), (e) and (g) (resp. (d),(f) and (h)) correspond to a simultaneous
fitting of the two decays and of the two saturation measurements for each pair
according to the four level model presented in the main text.

To investigate the system dynamics from a different perspective, we performed
saturation measurements under cw excitation as shown in Fig. 4 (g,h). The
saturation curves are well reproduced by the usual behavior $I\propto
P/(P+P_{\text{sat}})$, where the saturation power are $P_{sat,A_{1}}=14\pm
2\text{ mW}$, $P_{sat,A_{2}}=11\pm 2\text{ mW}$ and $P_{sat,B_{2}}=1.6\pm
0.3\text{ mW}$, whereas $P_{sat,B_{1}}$ could not be extracted for this range
of excitation densities. Within a pair, the ratio of the saturation power
should reflect the ratio of the decay rates of the corresponding levels. For
pair $B$, the ratio of $P_{\text{sat}}$ is at least larger than 15, thus
confirming the hypothesis that the lifetime of $B_{1}$ corresponds mainly to
the short component in the biexponential decay. In contrast emitters $A_{1}$
and $A_{2}$ show relatively similar saturation thresholds, indicating that
their unresolved rapid decay component are comparable.

We could reproduce simultaneously both the decay and the saturation curves of
each pair using the four-level scheme (Fig. 4 (a)). The green dashed lines for
pair $A$ and orange dashed lines for pair $B$ in Fig. 4 (c-h) correspond to a
simultaneous fitting of the two transients and of the two saturation curves
for each pair. We attempted to determine if the fast decay of the upper levels
could be related to an efficient tunnel coupling of the population to the
lower level in each pair (especially considering the small intra-pair OCC
distance of pair A ($\sim 1$ nm as estimated in the previous section, which is
comparable to the exciton Bohr radius in CNTs). This scenario is compatible
with the data. However, our limited time-resolution prevents us from observing
directly the rise time of the lower level signal that is expected in this
scenario.

##  4 Conclusion

We presented a thorough investigation of an original system composed of 4
OCCs, the emission energies of which are temporally correlated by pairs. Long
time-scale intensity auto-correlations show that the system cannot be
described with fewer emitters hosting several states. Using a full set of
optical tools, we showed that these emitters are most certainly all attached
to the same CNT. Using a thorough analysis and modeling of the correlated
Stark shifts, we could distinguish two closely packed pairs separated by
several hundreds of nm, while the distance between OCCs within a pair is on
the order of a few nm. Finally, the dynamics and saturation of the lines show
rather untypical behaviors for the upper levels with unusually fast decays,
however the resolution is not sufficient to prove unambiguously that this is
due to a population transfer to the lower levels; this hypothesis remains
however plausible, in view of the inter-OCC distances within a pair, which are
comparable to the exciton Bohr radius in nanotubes. This study reveals that
grafted OCCs on the backbone of a CNTs is a promising way to build ensembles
of closely packed quantum emitters, with possible electronic coupling. The
strange coincidence of finding two such pairs within a diffraction spot,
raises the question whether the grafting would be a collaborative process,
possibly fostered by the lattice deformation induced by the first OCC. The
question also comes if inter-level coupling could be achieved (at shorter
distance), paving the way to advanced quantum manipulations of coupled OCC
states.

#### Funding sources

This work was supported by the French National Research Agency (ANR) through
the grant DELICACY (ANR-22-CE47-0001) and by the FLAG-ERA Grant OPERA (DFG
437130745 and ANR-19-GRF1-0002).

The authors thanks X. He for sample preparation. The authors thank C.
Diederichs, E. Baudin, G. Hetet, Z. Said for fruitful discussion.

## References

  * Dolan et al. 2018 Dolan, P. R.; Adekanye, S.; Trichet, A. A. P.; Johnson, S.; Flatten, L. C.; Chen, Y. C.; Weng, L.; Hunger, D.; Chang, H.-C.; Castelletto, S.; Smith., J. M. Robust, tunable, and high purity triggered single photon source at room temperature using a nitrogen-vacancy defect in diamond in an open microcavity. _Opt. Express_ 2018, _26_ , 7056–7065
  * Wang et al. 2020 Wang, J.-F.; Yan, F.-F.; Li, Q.; Liu, Z.-H.; Liu, H.; Guo, G.-P.; Guo, L.-P.; Zhou, X.; Cui, J.-M.; Wang, J.; Zhou, Z.-Q.; Xu, X.-Y.; Xu, J.-S.; Li, C.-F.; Guo, G.-C. Coherent Control of Nitrogen-Vacancy Center Spins in Silicon Carbide at Room Temperature. _Phys. Rev. Lett._ 2020, _124_ , 223601
  * Dolde et al. 2013 Dolde, F.; Jakobi, I.; Naydenov, B.; Zhao, N.; Pezzagna, S.; Trautmann, C.; Meijer, J.; Neumann, P.; Jelezko, F.; Wrachtrup, J. Room-temperature entanglement between single defect spins in diamond. _Nature Physics_ 2013, _9_ , 139–143
  * Ji et al. 2024 Ji, W.; Liu, Z.; Guo, Y.; Hu, Z.; Zhou, J.; Dai, S.; Chen, Y.; Yu, P.; Wang, M.; Xia, K.; Shi, F.; Wang, Y.; Du, J. Correlated sensing with a solid-state quantum multisensor system for atomic-scale structural analysis. _Nature Photonics_ 2024,
  * Delord et al. 2024 Delord, T.; Monge, R.; Meriles, C. A. Correlated spectroscopy of stochastic Coulomb fields with sub-diffraction color center clusters. _arXiv Preprint_ 2024,
  * Koong et al. 2022 Koong, Z. X.; Cygorek, M.; Scerri, E.; Santana, T. S.; Park, S. I.; Song, J. D.; Gauger, E. M.; Gerardot, B. D. Coherence in cooperative photon emission from indistinguishable quantum emitters. _Science Advances_ 2022, _8_
  * Hettich et al. 2002 Hettich, C.; Schmitt, C.; Zitzmann, J.; Kühn, S.; Gerhardt, I.; Sandoghdar, V. Nanometer Resolution and Coherent Optical Dipole Coupling of Two Individual Molecules. _Science_ 2002, _298_ , 385–389
  * Trebbia et al. 2022 Trebbia, J.-B.; Deplano, Q.; Tamarat, P.; Lounis, B. Tailoring the superradiant and subradiant nature of two coherently coupled quantum emitters. _Nature Communications_ 2022, _13_
  * Piao et al. 2013 Piao, Y.; Meany, B.; Powell, L. R.; Valley, N.; Kwon, H.; Schatz, G. C.; Wang, Y. Brightening of carbon nanotube photoluminescence through the incorporation of sp3 defects. _Nat. Chem._ 2013, _5_ , 840–845
  * He et al. 2017 He, X.; Hartmann, N. F.; Ma, X.; Kim, Y.; Ihly, R.; Blackburn, J. L.; Gao, W.; Kono, J.; Yomogida, Y.; Hirano, A.; Tanaka, T.; Kataura, H.; Htoon, H.; Doorn, S. K. Tunable room-temperature single-photon emission at telecom wavelengths from sp3 defects in carbon nanotubes. _Nature Photonics_ 2017, _11_ , 577–582
  * Ma et al. 2015 Ma, X.; Hartmann, N. F.; Baldwin, J. K. S.; Doorn, S. K.; Htoon, H. Room-temperature single-photon generation from solitary dopants of carbon nanotubes. _Nature Nanotechnology_ 2015, _10_ , 671–675
  * Berger et al. 2019 Berger, F. J.; Lüttgens, J.; Nowack, T.; Kutsch, T.; Lindenthal, S.; Kistner, L.; Müller, C. C.; Bongartz, L. M.; Lumsargis, V. A.; Zakharko, Y.; Zaumseil, J. Brightening of Long, Polymer-Wrapped Carbon Nanotubes by sp3 Functionalization in Organic Solvents. _ACS Nano_ 2019, _13_ , 9259–9269
  * Settele et al. 2021 Settele, S.; Berger, F. J.; Lindenthal, S.; Zhao, S.; Yumin, A. A. E.; Zorn, N. F.; Asyuda, A.; Zharnikov, M.; Högele, A.; Zaumseil, J. Synthetic control over the binding configuration of luminescent sp3-defects in single-walled carbon nanotubes. _Nature Communications_ 2021, _12_
  * Zheng et al. 2021 Zheng, Y.; Kim, Y.; Jones, A. C.; Olinger, G.; Bittner, E. R.; Bachilo, S. M.; Doorn, S. K.; Weisman, R. B.; Piryatinski, A.; Htoon, H. Quantum Light Emission from Coupled Defect States in DNA-Functionalized Carbon Nanotubes. _ACS Nano_ 2021, _15_ , 10406–10414, Publisher: American Chemical Society
  * Jeantet et al. 2016 Jeantet, A.; Chassagneux, Y.; Raynaud, C.; Roussignol, P.; Lauret, J.; Besga, B.; Estève, J.; Reichel, J.; Voisin, C. Widely Tunable Single-Photon Source from a Carbon Nanotube in the Purcell Regime. _Physical Review Letters_ 2016, _116_
  * Borel et al. 2023 Borel, A.; Habrant-Claude, T.; Rapisarda, F.; Reichel, J.; Doorn, S. K.; Voisin, C.; Chassagneux, Y. Telecom Band Single-Photon Source Using a Grafted Carbon Nanotube Coupled to a Fiber Fabry–Perot Cavity in the Purcell Regime. _ACS Photonics_ 2023, _10_ , 2839–2845
  * Gandil et al. 2019 Gandil, M.; Matsuda, K.; Lounis, B.; Tamarat, P. Spectroscopic signatures of spin-orbit coupling and free excitons in individual suspended carbon nanotubes. _Phys. Rev. B_ 2019, _100_ , 081411
  * Kim et al. 2020 Kim, Y.; Goupalov, S. V.; Weight, B. M.; Gifford, B. J.; He, X.; Saha, A.; Kim, M.; Ao, G.; Wang, Y.; Zheng, M.; Tretiak, S.; Doorn, S. K.; Htoon, H. Hidden Fine Structure of Quantum Defects Revealed by Single Carbon Nanotube Magneto-Photoluminescence. _ACS Nano_ 2020, _14_ , 3451–3460, _eprint: https://doi.org/10.1021/acsnano.9b09548
  * Boiron et al. 1999 Boiron, A.-M.; Tamarat, P.; Lounis, B.; Brown, R.; Orrit, M. Are the spectral trails of single molecules consistent with the standard two-level system model of glasses at low temperatures? _Chemical Physics_ 1999, _247_ , 119–132
  * Roquelet et al. 2012 Roquelet, C.; Vialla, F.; Diederichs, C.; Roussignol, P.; Delalande, C.; Deleporte, E.; Lauret, J.-S.; Voisin, C. Local Field Effects in the Energy Transfer between a Chromophore and a Carbon Nanotube: A Single-Nanocompound Investigation. _ACS Nano_ 2012, _6_ , 8796–8802
  * Raynaud et al. 2019 Raynaud, C.; Claude, T.; Borel, A.; Amara, M. R.; Graf, A.; Zaumseil, J.; Lauret, J.-S.; Chassagneux, Y.; Voisin, C. Superlocalization of Excitons in Carbon Nanotubes at Cryogenic Temperature. _Nano Letters_ 2019, _19_ , 7210–7216, PMID: 31487461
  * Danné et al. 2018 Danné, N.; Kim, M.; Godin, A. G.; Kwon, H.; Gao, Z.; Wu, X.; Hartmann, N. F.; Doorn, S. K.; Lounis, B.; Wang, Y.; Cognet, L. Ultrashort Carbon Nanotubes That Fluoresce Brightly in the Near-Infrared. _ACS Nano_ 2018, _12_ , 6059–6065, PMID: 29889499
  * Glückert et al. 2018 Glückert, J. T.; Adamska, L.; Schinner, W.; Hofmann, M. S.; Doorn, S. K.; Tretiak, S.; Högele, A. Dipolar and charged localized excitons in carbon nanotubes. _Phys. Rev. B_ 2018, _98_ , 195413
  * Zheng et al. 2021 Zheng, Y.; Weight, B. M.; Jones, A. C.; Chandrasekaran, V.; Gifford, B. J.; Tretiak, S.; Doorn, S. K.; Htoon, H. Photoluminescence Dynamics Defined by Exciton Trapping Potential of Coupled Defect States in DNA-Functionalized Carbon Nanotubes. _ACS Nano_ 2021, _15_ , 923–933, PMID: 33395262
  * Kim et al. 2019 Kim, Y.; Velizhanin, K. A.; He, X.; Sarpkaya, I.; Yomogida, Y.; Tanaka, T.; Kataura, H.; Doorn, S. K.; Htoon, H. Photoluminescence Intensity Fluctuations and Temperature-Dependent Decay Dynamics of Individual Carbon Nanotube sp3 Defects. _J. Phys. Chem. Lett._ 2019, _10_ , 1423–1430, Publisher: American Chemical Society

##  5 Supplementary informations

##  6 Temporal correlations

###  6.1 Grafted SWCNT

We recorded a series of 2500 spectra each with an integration time of 1s (time
traces) for several excitation powers. We fitted each emission line by a
Lorentzian function and extracted their emission energies and intensities. We
computed the Pearson correlation coefficient between these parameters. For the
sake of readability, we artificially set the auto-correlation coefficients to
0. The spectral fluctuations increase with excitation power density, pointing
to a photo-induced effect, possibly mediated by photo-generated charges.

Figure S-5: Temporal correlations between emission energies for pair A and B
for different excitation power. For each power, the time trace is displayed as
a map in gray color on a reduced scale. Energy deviation to the mean value
$\Delta E(t)=E(t)-<E>$ is then displayed for both pair (coloured 2D plot)
before computing the Pearson correlation coefficients.

Figure S-6: Temporal correlations between intensities for pair A and B for
different excitation power. For each power, the time trace is displayed as a
map in gray color on a reduced scale. Intensity deviation to the mean value
$\Delta I(t)=I(t)-<I>$ is then displayed for both pair (coloured 2D plot)
before computing the Pearson correlation coefficients. The data show no
significant correlation between the intensities of the peaks neither within a
pair nor between them.

Figure S-7: Intensity to emission energy Pearson correlations for different
excitation power. The data show no significant correlations.

##  7 Super-localization

The super-localization technique is used to locate a nano-object with a
precision much below the diffraction limit. It consists in measuring the PL
intensity while scanning the sample spatially in a confocal setup. The
resulting map is the convolution of the emitter geometry with the point spread
function (PSF) of the optical setup. For a point-like emitter (such as a nano-
emitter) the PL map boils down to the PSF. This latter can be modeled by a 2D
Gaussian function, which in the appropriate (eigen) frame, reads:

$S(x,y)=A\exp(-\tfrac{(x-x_{0})^{2}}{2\sigma_{x}^{2}})\exp(-\tfrac{(y-y_{0})^{2}}{2\sigma_{y}^{2}})+B,$
(2)

where $A$ is the amplitude, $x_{0}$ and $y_{0}$ are the spatial coordinates of
the emitter, $\sigma_{x,y}$ is the width of the Gaussian function in the $x$
and $y$ directions respectively and $B$ is a global offset.

In our experimental setup we coupled the emitter PL into a single mode fiber.
The fiber core acts as a $10\mu m$ pinhole ensuring spatial selectivity. The
emitters are excited from the backside of the substrate with a cw 800nm laser
and the PL is collected through an aspheric lens (Thorlabs C330-TND-C,
NA=0.7). The core of the fiber is thus conjugated to the collection spot on
the sample which can be scanned using a fast steering mirror in the 4f
configuration.

To get the best of the superlocalization technique, the measurement should
last less than the typical mechanical drift. Due to the low signal level, we
chose to work with a 150 lines/mm grating to reduce the integration time down
to 7s per spectra, for a typical mapping time of 15 minutes. Thus, only the
two brightest emitters, A1 and B2, could be resolved (Figure S-8(a)).

The spectrally resolved map of emitter A1 (resp. B2) is displayed in Figure
S-8.(c) (resp. S-8(d)). These maps were fitted by a 2D Gaussian function as
shown in figure S-8.(e-f). The center of the Gaussian is reported in map (g).
The measurement was repeated 4 times successively to assess the mechanical
drift. Each measurement is identified by a different symbol among which the
color circle (resp. diamond) symbol stands for the result of the first (resp.
last) measurement. We can observe for both emitter a typical drift of 200 nm
in the Y direction between the first and last measurements. The barycentres
are represented in black. We end up with typical distance of 260 $\pm$ 100 nm
between the emitters, which allow us to unambiguously confirm that the two
pairs A and B are well separated and can be considered as independent.

Figure S-8: Superlocalization of emitter A1 and B2 using a single mode fiber
as a pinhole. (a) Typical spectrum recorded with a 150 lines/mm grating. (c-d)
Spectrally resolved PL maps of emitters A1 and B2. (e,f) 2D Gaussian fits of
the maps in (c) and (d). (g) Centers of the Gaussian fits, representing the
super-localisation of the emitters A1 and B2 for 4 acquisitions realized in a
row. The barycenter is figured in black. Error bars resulting from the
Gaussian fits.

##  8 Couple B: possible 3D electrostatic configuration

In Figure S-9, some possible 3D configurations of the dipoles associated with
pair B are presented, that are compatible with the experimental data presented
in Figure 3 of the main text. The average slope of the central lobe (Z1)
(corresponding to trapping and detrapping of remote charges) sets the ratio of
the projections of the two dipoles along the nanotube axis. The lobes Z2 and
Z3 correspond to charges in the vicinity of the dipoles, for which the 3D
geometry matters. We search numerically for such a 3D geometry implying the
orientation and position of the two dipoles and the position of the two charge
traps responsible for the lobes Z2 and Z3. A range of possible configurations
are found among which we show here six examples.

Figure S-9: Six examples of possible 3D configurations compatible with the
spectral diffusion observed in pair B. The green arrow indicates the permanent
dipole. The blue (resp. magenta) dot represents the position of the trap
corresponding to zone Z2 (resp. zone Z3) of Figure 3 of the main text.

To further confirm the adequacy of such a configuration with the whole set of
experimental data, we try to reproduce directly all the experimental points
observed in Figure 3 of the main paper. To do so, for each of the
configurations found in fig S-9, we search numerically for the positions of
the trapping or detrapping events responsible for the main lobe Z1, using trap
positions as close as possible to the nanotube surface (as an example see
Figure S-10, corresponding to the third dipole configuration displayed in
Figure S-9).

Figure S-10: Each blue or red dots correspond to an experimental point of
fig3c of the main text.

##  9 Autocorrelation function for a fast switching emitter

In this paragraph we compute the auto-correlation function of a two-level
system emitter that switches between two configurations, resulting in two
distinct emission lines when observed with a long integration time. The
switching rates between the two configurations (labelled ”1” and ”2”) are
given by $\gamma_{12}$ and $\gamma_{21}$.

The autocorrelation function is given by

$g^{(2)}_{state=i}(\tau)=\frac{\langle I_{i}(t+\tau)I_{i}(t)\rangle}{\langle
I_{i}(t)\rangle^{2}},$ (3)

where $i=1,2$ corresponds to the configuration. The autocorrelation can be
related to conditional probabilities:

$g^{(2)}_{state=1}(\tau)=\frac{\Big{(}P(state=1\ @t+\tau)|state=1\
@t)\Big{)}P(state=1\ @t)}{P(state=1\ @t)^{2}}.$ (4)

The probabilities are given by ($\tau>0$):

$\displaystyle P(state=1\ @t+\tau)|state=1\ @t)$ $\displaystyle=$
$\displaystyle\eta+(1-\eta)e^{-\gamma\tau},$ (5) $\displaystyle P(state=2\
@t+\tau)|state=2\ @t)$ $\displaystyle=$ $\displaystyle(1-\eta)+\eta
e^{-\gamma\tau},$ (6) $\displaystyle P(state=1\ @t)$ $\displaystyle=$
$\displaystyle\eta,$ (7) $\displaystyle P(state=2\ @t)$ $\displaystyle=$
$\displaystyle 1-\eta,$ (8)

where $\eta$ and $\gamma$ are defined by $\gamma_{21}=\eta\gamma$,
$\gamma_{12}=(1-\eta)\gamma$. This leads to:

$\displaystyle g^{(2)}_{state=1}(\tau)$ $\displaystyle=$ $\displaystyle
1+\frac{1-\eta}{\eta}e^{-\gamma|\tau|},$ (9) $\displaystyle
g^{(2)}_{state=2}(\tau)$ $\displaystyle=$ $\displaystyle
1+\frac{\eta}{1-\eta}e^{-\gamma|\tau|}.$ (10)

A fast switching event is thus characterized by a step (in semi-logx scale) in
the auto-correlation of both lines, at the same characteristic time
$1/\gamma$. Moreover the product of the amplitudes of this step of each line
is equal to one (for a normalized $g^{(2)}$ function).

##  10 1D vs 3D emitter pair generation probability

For an emitter numeric concentration $\rho$, the probability to find $k$
emitters within an elementary volume $v$ reads:
$P_{k}(v)=e^{-v\rho}(v\rho)^{k}/k!$. The manipulation of a quantum coupled
pair requires that two emitters are in close vicinity (within $dv~{}$ a few
nm3) and are the only pair within a volume $V_{\lambda}$ of the order of
$\lambda^{3}$, where $\lambda$ is the transition wavelength used to manipulate
the system. This configuration has a probability
$P=\frac{V_{\lambda}}{dv}P_{2}(dv)P_{0}(V_{\lambda}-dv)$. This probability is
maximum for a concentration $\rho=2/V_{\lambda}$ and amounts to
$P_{max}=\frac{dv}{V_{\lambda}}2e^{-2}$. In a 3D system, with $dv=\delta^{3}$,
this reads: $P_{max}\simeq 0.27(\delta/\lambda)^{3}$, whereas for a 1D system,
one gets $P_{max}\simeq 0.27\delta/\lambda$. Hence, for typical configurations
where $\delta/\lambda\simeq 0.001$, the 1D configuration is $10^{6}$ times
more favorable than its 3D counterpart, showing the strong interest of using a
nanotube as a 1D backbone to anchor the OCCs.


# Controlling entanglement sudden death and birth in cavity QED

Jian-Song Zhang  Jing-Bo Xu<EMAIL_ADDRESS>Zhejiang Institute of Modern
Physics and Physics Department,
Zhejiang University, Hangzhou 310027, China

###### Abstract

We present a scheme to control the entanglement sudden birth and death in
cavity quantum electrodynamics system, which consists of two noninteracting
atoms each locally interacting with its own vacuum field, by applying and
adjusting classical driving fields.

###### pacs:

03.67.Mn; 03.65.Ud

##  I INTRODUCTION

In recent years, entanglement has been considered as a key resource of quantum
information processing 1 ; 2 ; 3 ; 4 . A cavity quantum electrodynamics(QED)
system is a useful tool to create the entanglement between atoms in cavities
and establish quantum communications between different optical cavities.
Recently, the manipulation of quantum entanglement for the system of cavity
QED has been extensively investigated5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 .

Many efforts have been devoted to the study of the evolution of the
entanglement under the influence of the environment 14 ; 15 ; 16 ; Yu2009 ; 17
; 18 ; 19 . It is pointed out by Yu and Eberly 14  that the entanglement of an
entangled two-qubit interacting with uncorrelated reservoirs may disappear
within a finite time during the dynamics evolution. This phenomenon, called
entanglement sudden death (ESD) has been observed in experiment 20 ; 21 .
Recently, the entanglement sudden birth(ESB) in cavity QED has been discussed
by Yönac, Yu, and Eberly Yonac2006 ; Yonac2007 . More recently, Lopez _et al_.
22  have studied the entanglement dynamics of a quantum system consisting of
two cavities interacting with two independent reservoirs and shown that ESD in
a bipartite system independently coupled to reservoirs is related to the ESB.
It has been pointed out that the cavity coherent state can be used to control
the ESB and ESD in cavity QEDYonac2008 .

In the present paper, we propose a scheme to control ESB and ESD of a quantum
system consisting of two noninteracting atoms each locally interacting with
its own vacuum field. The two atoms, which are initially prepared in entangled
states, are driven by two classical fields additionally. It is shown that ESB
and ESD phenomenon may appear in this system and the time of ESB and ESD can
be controlled by classical driving fields. In addition, the amount of the
entanglement of the two atoms or cavities can be significantly increased by
applying classical fields.

##  II Effective Hamiltonian

Now, we consider a system consisting of a two-level atom inside a single mode
cavity. The atom is driven by a classical field additionally. The Hamiltonian
of the system can be described by 12

$\displaystyle H$ $\displaystyle=$ $\displaystyle\omega
a^{{\dagger}}a+\frac{\omega_{0}}{2}\sigma_{z}+g(\sigma_{+}a+\sigma_{-}a^{{\dagger}})$
(1)
$\displaystyle+\lambda(e^{-i\omega_{c}t}\sigma_{+}+e^{i\omega_{c}t}\sigma_{-}),$

where $\omega$, $\omega_{0}$ and $\omega_{c}$ are the frequency of the cavity,
atom and classical field, respectively. The operators $\sigma_{z}$ and
$\sigma_{\pm}$ are defined by $\sigma_{z}=|e\rangle\langle e|-|g\rangle\langle
g|$, $\sigma_{+}=|e\rangle\langle g|$, and $\sigma_{-}=\sigma_{+}^{{\dagger}}$
where $|e\rangle$ and $|g\rangle$ are the excited and ground states of the
atom. Here, $a$ and $a^{{\dagger}}$ are the annihilation and creation
operators of the cavity; g and $\lambda$ are the coupling constants of the
interactions of the atom with the cavity and with the classical driving field,
respectively. Note that we have set $\hbar=1$ throughout this paper.

In the rotating reference frame the Hamiltonian of the system is transformed
to the Hamiltonian $H_{1}$ under a unitary transformation
$U_{1}=\exp{(-i\omega_{c}t\sigma_{z}/2)}$

$\displaystyle H_{1}$ $\displaystyle=$ $\displaystyle
U_{1}^{{\dagger}}HU_{1}-iU_{1}^{{\dagger}}\frac{\partial U_{1}}{\partial t}$
(2) $\displaystyle=$ $\displaystyle H_{1}^{(1)}+H_{1}^{(2)},$

with

$\displaystyle H_{1}^{(1)}$ $\displaystyle=$ $\displaystyle\omega
a^{{\dagger}}a+g(e^{i\omega_{c}t}\sigma_{+}a+e^{-i\omega_{c}t}\sigma_{-}a^{{\dagger}}),$
$\displaystyle H_{1}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{\Delta_{1}}{2}\sigma_{z}+\lambda(\sigma_{+}+\sigma_{-}),$
(3)

and $\Delta_{1}=\omega_{0}-\omega_{c}$. Using the method similar to that used
in Ref.23 , diagonalizing the Hamiltonian $H_{1}^{(2)}$, and neglecting the
terms which do not conserve energies (rotating wave approximation), we can
recast the Hamiltonian $H_{1}$ as follows:

$\displaystyle H_{1}$ $\displaystyle=$ $\displaystyle\omega
a^{{\dagger}}a+\frac{\Omega_{1}\sin{\theta}}{2}(\sigma_{+}+\sigma_{-})+g\cos^{2}{\frac{\theta}{2}}[e^{i\omega_{c}t}$
(4)
$\displaystyle\times(-\frac{\sin{\theta}}{2}\sigma_{z}+\cos^{2}{\frac{\theta}{2}}\sigma_{+}-\sin^{2}{\frac{\theta}{2}}\sigma_{-})a+h.c],$

with $\theta=\arctan{(\frac{2\lambda}{\Delta_{1}})}$. Here $h.c$ stands for
Hermitian conjugation.

The Hamiltonian $H_{1}$ can be diagonalized by a final unitary transformation
$U_{2}$ with $U_{2}=\exp{[\frac{i\omega_{c}t}{2}(\sigma_{+}+\sigma_{-})]}$.
Then, we can rewrite the Hamiltonian of the system

$\displaystyle H_{2}$ $\displaystyle=$ $\displaystyle\omega
a^{{\dagger}}a+\frac{\omega^{\prime}\sin{\theta}}{2}(\sigma_{+}+\sigma_{-})+g^{\prime}[(-\frac{\sin{\theta}}{2}\sigma_{z}$
(5)
$\displaystyle+\cos^{2}{\frac{\theta}{2}}\sigma_{+}-\sin^{2}{\frac{\theta}{2}}\sigma_{-})a+h.c],$

where $\omega^{\prime}=\sqrt{\Delta_{1}^{2}+4\lambda^{2}}+\omega_{c}$ and
$g^{\prime}=g\cos^{2}{\frac{\theta}{2}}$. It is worth noting that the unitary
transformations $U_{1}$ and $U_{2}$ are both local unitary transformations. As
we known the entanglement of a quantum system does not change under local
unitary transformations 24 . Thus, the entanglement of the system considered
here will not be changed by applying the local unitary transformations $U_{1}$
and $U_{2}$.

##  III Controlling entanglement sudden death and birth

In this section, we investigate ESD and ESB of a quantum system consisting of
two noninteracting atoms each locally interacting with its own vacuum field.
Each atom interacts with its own vacuum field where the interaction of the
system is described by $H_{2}$. We show how to control entanglement sudden
death and birth of a quantum system formed by two two-level atoms and two
cavities via classical driving fields. Assume the two-level atoms are prepared
in entangled states and the cavities are prepared in vacuum states, i.e., the
whole system is initially prepared in the state

$\displaystyle|\psi(0)\rangle=(\alpha|-_{a_{1}}\rangle|-_{a_{2}}\rangle+\beta|+_{a_{1}}\rangle|+_{a_{2}}\rangle)|0_{c_{1}}\rangle|0_{c_{2}}\rangle,$
(6)

where the subscripts $a_{1}$, $a_{2}$, $c_{1}$, and $c_{2}$ refer to atom 1,
atom 2, cavity 1, and cavity 2, respectively. Here, $|\pm\rangle$ can be
interpreted as the dressed states of the two-level atom. They are defined as
follows:

$\displaystyle|+\rangle$ $\displaystyle=$
$\displaystyle\cos{\frac{\theta}{2}}|e\rangle+\sin{\frac{\theta}{2}}|g\rangle,$
$\displaystyle|-\rangle$ $\displaystyle=$
$\displaystyle-\sin{\frac{\theta}{2}}|e\rangle+\cos{\frac{\theta}{2}}|g\rangle.$
(7)

After some algebra, we find the state of the whole system at time t is

$\displaystyle|\psi(t)\rangle$ $\displaystyle=$
$\displaystyle\alpha|-_{a_{1}}\rangle|-_{a_{2}}\rangle|0_{c_{1}}\rangle|0_{c_{2}}\rangle$
(8) $\displaystyle+\beta
f^{2}_{1}(t)|+_{a_{1}}\rangle|+_{a_{2}}\rangle|0_{c_{1}}\rangle|0_{c_{2}}\rangle$
$\displaystyle+\beta
f^{2}_{2}(t)|-_{a_{1}}\rangle|-_{a_{2}}\rangle|1_{c_{1}}\rangle|1_{c_{2}}\rangle$
$\displaystyle+\beta
f_{1}(t)f_{2}(t)(|+_{a_{1}}\rangle|-_{a_{2}}\rangle|0_{c_{1}}\rangle|1_{c_{2}}\rangle$
$\displaystyle+|-_{a_{1}}\rangle|+_{a_{2}}\rangle|1_{c_{1}}\rangle|0_{c_{2}}\rangle),$

with

$\displaystyle f_{1}(t)$ $\displaystyle=$ $\displaystyle
e^{i\Delta_{2}t/2}[\cos{(\Omega t)}-\frac{i\Delta_{2}}{2\Omega}\sin{(\Omega
t)}],$ $\displaystyle f_{2}(t)$ $\displaystyle=$ $\displaystyle-
ig\cos^{2}{\frac{\theta}{2}}e^{-i\Delta_{2}t/2}\sin{(\Omega t)}/\Omega,$
$\displaystyle\Delta_{2}$ $\displaystyle=$
$\displaystyle\sqrt{(\omega_{0}-\omega_{c})^{2}+4\lambda^{2}}+\omega_{c}-\omega,$
$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\Delta_{2}^{2}}{4}+(g\cos^{2}{\frac{\theta}{2}})^{2}}.$
(9)

Tracing over the degrees of the freedom of cavities, we obtain the reduced
density matrix of two atoms

$\displaystyle\rho_{a_{1}a_{2}}(t)$ $\displaystyle=$
$\displaystyle[|\alpha|^{2}+|\beta
f^{2}_{2}(t)|^{2}]|-_{a_{1}}\rangle|-_{a_{2}}\rangle\langle-_{a_{1}}|\langle-_{a_{2}}|$
(10) $\displaystyle+|\beta
f^{2}_{1}(t)|^{2}|+_{a_{1}}\rangle|+_{a_{2}}\rangle\langle+_{a_{1}}|\langle+_{a_{2}}|$
$\displaystyle+|\beta
f_{1}(t)f_{2}(t)|^{2}(|+_{a_{1}}\rangle|-_{a_{2}}\rangle\langle+_{a_{1}}|\langle-_{a_{2}}|$
$\displaystyle+|-_{a_{1}}\rangle|+_{a_{2}}\rangle\langle-_{a_{1}}|\langle+_{a_{2}})$
$\displaystyle+[\alpha\beta^{*}f_{1}^{*2}(t)|-_{a_{1}}\rangle|-_{a_{2}}\rangle\langle+_{a_{1}}|\langle+_{a_{2}}|+h.c].$

Similarly, the reduced density matrix of two cavities is

$\displaystyle\rho_{c_{1}c_{2}}(t)$ $\displaystyle=$
$\displaystyle[|\alpha|^{2}+|\beta
f^{2}_{1}(t)|^{2}]|-_{a_{1}}\rangle|-_{a_{2}}\rangle\langle-_{a_{1}}|\langle-_{a_{2}}|$
(11) $\displaystyle+|\beta
f^{2}_{2}(t)|^{2}|+_{a_{1}}\rangle|+_{a_{2}}\rangle\langle+_{a_{1}}|\langle+_{a_{2}}|$
$\displaystyle+|\beta
f_{1}(t)f_{2}(t)|^{2}(|+_{a_{1}}\rangle|-_{a_{2}}\rangle\langle+_{a_{1}}|\langle-_{a_{2}}|$
$\displaystyle+|-_{a_{1}}\rangle|+_{a_{2}}\rangle\langle-_{a_{1}}|\langle+_{a_{2}})$
$\displaystyle+[\alpha\beta^{*}f_{2}^{*2}(t)|-_{a_{1}}\rangle|-_{a_{2}}\rangle\langle+_{a_{1}}|\langle+_{a_{2}}|+h.c].$

In order to study the entanglement of above system described by density matrix
$\rho$, we adopt the measure concurrence which is defined as 25

$C=\max{\\{0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\\}},$ (12)

where the $\lambda_{i}$(i=1,2,3,4) are the square roots of the eigenvalues in
decreasing order of the magnitude of the “spin-flipped” density matrix
operator
$R=\rho(\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y})$ and
$\sigma_{y}$ is the Pauli Y matrix, i.e.,
$\sigma_{y}=\left(\begin{array}[]{cc}0&-i\\\ i&0\end{array}\right)$.
Particularly, for a density matrix of the form

$\displaystyle\rho=\left(\begin{array}[]{cccc}a&0&0&0\\\ 0&b&z&0\\\
0&z^{*}&c&0\\\ 0&0&0&d\end{array}\right),$ (17)

the concurrence is

$\displaystyle C=2\max\\{0,|z|-\sqrt{ad}\\}.$ (18)

Combing the above equation with the reduced density matrix, we find that the
concurrence of two atoms is

$\displaystyle C_{a_{1}a_{2}}(t)=2|f_{1}(t)|^{2}\max\\{0,|\alpha\beta|-|\beta
f_{2}(t)|^{2}\\},$ (19)

and the concurrence of two cavities is

$\displaystyle C_{c_{1}c_{2}}(t)=2|f_{2}(t)|^{2}\max\\{0,|\alpha\beta|-|\beta
f_{1}(t)|^{2}\\}.$ (20)

In Fig.1, the evolution of two-qubit concurrence for different partitions
$C_{a_{1}a_{2}}$ (solid line) and $C_{c_{1}c_{2}}$ (dotted line) are plotted
with $\alpha=1/\sqrt{10},\beta=3/\sqrt{10},\omega=3,\omega_{0}=2,g=1$. For
simplicity, we sometimes choose the special case of
$\omega:\omega_{0}:\omega_{c}=3:2:1$. On the one hand, the concurrence of two
atoms $C_{a_{1}a_{2}}$ will disappear within a finite time during the dynamics
evolution(ESD). On the other hand, the concurrence of two cavities
$C_{c_{1}c_{2}}$ can appear during the dynamics evolution(see the dotted line
in Fig.1). It is not difficult to see that the time for which ESD($t_{ESD}$)
and ESB($t_{ESB}$) occur could be adjusted by controlling the frequency
$\omega_{c}$ and strength $\lambda$ of classical driving fields. In addition,
the amount of entanglement between two cavities can also be controlled by
classical driving fields.

In order to show this more clearly, we plot the two-qubit concurrence for
different partitions $C_{a_{1}a_{2}}$ (solid line) and $C_{c_{1}c_{2}}$
(dotted line) with
$\alpha=\sqrt{3}/\sqrt{10},\beta=\sqrt{7}/\sqrt{10},\omega=3,\omega_{0}=2,g=1$
in Fig.2. Comparing Fig.1 and Fig.2, one can see time of ESD($t_{ESD}$) and
ESB($t_{ESB}$) depend on the parameters $\alpha$ and $\beta$. In the case of
$\alpha=1/\sqrt{10}$ and $\beta=3/\sqrt{10}$, $t_{ESD}<t_{ESB}$, that is, ESB
appears after ESD. However, when $\alpha=\sqrt{3}/\sqrt{10}$ and
$\beta=\sqrt{7}/\sqrt{10}$, $t_{ESD}>t_{ESB}$, that is, ESB appears before
ESD. Again, the time of ESD and ESB and the amount of entanglement between two
cavities can be controlled by adjusting classical driving fields.

We now turn to show the influence of classical driving fields on the
distribution of entanglement in the present system. The bipartite entanglement
of $a_{1}\otimes a_{2}$, $c_{1}\otimes c_{2}$, $a_{1}\otimes c_{2}$, and
$c_{1}\otimes a_{2}$ are displayed in Fig.3. It is not difficult to see that
the concurrence $C_{a_{1}a_{2}}$, $C_{c_{1}c_{2}}$, $C_{a_{1}c_{2}}$, and
$C_{c_{1}a_{2}}$ are periodic functions of time t. The periods of them depend
on the strength and the frequencies of classical driving fields. Comparing the
right panel and the left panel of Fig.3, we find that the time of ESB and ESD
and the amount of the entanglement of two qubits can be controlled by
classical driving fields. For example, $t_{ESD}$ and the amount of
$C_{c_{1}c_{2}}$(dashed line) of the right panel are larger than that of the
left panel.

##  IV CONCLUSIONS

In summary, we have considered a quantum system consisting of two
noninteracting atoms each locally interacting with its own vacuum field. The
two atoms, which are driven by two classical fields, are initially prepared in
entangled states. We find that classical driving fields can increase the
amount of entanglement of the two-atom system. It is worth noting that the
time of ESB and ESD can be controlled by the classical driving fields. The
approach presented in the present Letter may have potential applications in
quantum information processing.

##  ACKNOWLEDGEMENTS

This project was supported by the National Natural Science Foundation of China
(Grant No.10774131) and the National Key Project for Fundamental Research of
China (Grant No. 2006CB921403).

## References

  * (1) J. I. Cirac, P. Zoller, Nature 404 (2000) 579.
  * (2) M.A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).
  * (3) C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. K. Wootters, Phys. Rev. Lett. 70 (1993) 1895\.
  * (4) V. Vedral, M. B. Plenio, Phys. Rev. A 57 (2000) 1619.
  * (5) J. M. Raimond, M. Brune, S. Haroche, Rev. Mod. Phys 73 (2001) 565.
  * (6) S. Bose, I. Fuentes-Guridi, P. L. Knight, V. Vedral, Phys. Rev. Lett. 87 (2001) 050401.
  * (7) M. S. Kim, Jinhyoung Lee, D. Ahn, P. L. Knight, Phys. Rev. A 65 (2002) 040101(R).
  * (8) M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, M. Aspelmeyer, Phys. Rev. Lett. 99 (2007) 250401.
  * (9) D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, M. Aspelmeyer, Phys. Rev. Lett. 98 (2007) 030405.
  * (10) S. B. Li and J. B. Xu, Phys. Rev. A 72 (2005) 022332.
  * (11) J. B. Xu and S. B. Li, New. J. Phys. 7 (2005) 72.
  * (12) E. Solano, G. S. Agarwal, and H. Walther, Phys. Rev. Lett. 90  (2003) 027903\.
  * (13) S. B. Li and J. B. Xu, Phys. Lett. A 337 (2005) 321\.
  * (14) Y. Yu and J. H. Eberly, Phys. Rev. Lett. 93 (2004) 140404; Y. Yu and J. H. Eberly, Opt. Commun. 264 (2006) 393.
  * (15) Z. Ficek and R. Tanaś, Rev. Rev. A 74 (2006) 024304.
  * (16) T. Yu and J. H. Eberly, Quantum Inf. Comput. 7 (2007) 459.
  * (17) T. Yu and J. H. Eberly, Science(London) 323 (2009) 598\.
  * (18) M. Ikram, Fu-Li Li, and M. S. Zubairy, Phys. Rev. A 75 (2007) 062336.
  * (19) B. Bellomo, R. Lo France, and G. Compagno, Phys. Rev. A 77 (2008) 032342.
  * (20) A. R. P. Rau, Mazhar Ali and G. Alber, Europhys. Lett. 82 (2008) 40002; J.-S. Zhang, J.-B. Xu, Opt. Commun. (2009), doi:10.1016/j.optcom.2009.03.015.
  * (21) M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, L. Davidovich, Science(London) 316 (2007) 579\.
  * (22) J. Laurat, K. S. Choi, H. Deng, C. W. Chou, and H. J. Kimble, Phys. Rev. Lett. 99 (2007) 180504.
  * (23) M. Yönac, T. Yu, J. H. Eberly, J. Phys. B 39 (2006) S621 .
  * (24) M. Yönac, T. Yu, J. H. Eberly, J. Phys. B 40 (2007) S45.
  * (25) C. E. Lopez, G. Romero, F. Lastra, E. Solano, and J. C. Retamal, Phys. Rev. Lett. 101 (2008) 080503.
  * (26) M. Yönac and J.H. Eberly, Opt. Lett. 33 (2008) 270.
  * (27) Yu-Xi Liu, C. P. Sun, and Franco Nori, Phys. Rev. A 74 (2006) 052321.
  * (28) V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Phys. Rev. Lett. 78 (1997) 2275\.
  * (29) W. K. Wootters, Phys. Rev. Lett. 80 (1998) 2245.

List of captions

FIG.1 The concurrence of two atoms (solid line) and two caviteis (dotted line)
are plotted as a function of t with
$\alpha=1/\sqrt{10},\beta=3/\sqrt{10},\omega=3,\omega_{0}=2,g=1$. Right panel:
$\omega_{c}=\lambda=0$. Left panel: $\omega_{c}=\lambda=1$.

FIG.2 The concurrence of two atoms (solid line) and two caviteis (dotted line)
are plotted as a function of t with
$\alpha=\sqrt{3}/\sqrt{10},\beta=\sqrt{7}/\sqrt{10},\omega=3,\omega_{0}=2,g=1$.
Right panel: $\omega_{c}=\lambda=0$. Left panel: $\omega_{c}=\lambda=1$.

FIG.3 The concurrence of two qubits for different partitions are plotted as a
function of t with
$\alpha=\sqrt{3}/\sqrt{10},\beta=\sqrt{7}/\sqrt{10},\omega=3,\omega_{0}=2,g=1$.
Right panel: $\omega_{c}=\lambda=0$. Left panel: $\omega_{c}=\lambda=1$.

Figure 1: The concurrence of two atoms (solid line) and two caviteis (dotted
line) are plotted as a function of t with
$\alpha=1/\sqrt{10},\beta=3/\sqrt{10},\omega=3,\omega_{0}=2,g=1$. Right panel:
$\omega_{c}=\lambda=0$. Left panel: $\omega_{c}=\lambda=1$.

Figure 2: The concurrence of two atoms (solid line) and two caviteis (dotted
line) are plotted as a function of t with
$\alpha=\sqrt{3}/\sqrt{10},\beta=\sqrt{7}/\sqrt{10},\omega=3,\omega_{0}=2,g=1$.
Right panel: $\omega_{c}=\lambda=0$. Left panel: $\omega_{c}=\lambda=1$.

Figure 3: The concurrence of two qubits for different partitions are plotted
as a function of t with
$\alpha=\sqrt{3}/\sqrt{10},\beta=\sqrt{7}/\sqrt{10},\omega=3,\omega_{0}=2,g=1$.
Right panel: $\omega_{c}=\lambda=0$. Left panel: $\omega_{c}=\lambda=1$.