Quantum coherence effects in hybrid
nanoparticle molecules in the presence of
ultra-short dephasing times
Cite as: Appl. Phys. Lett. 101, 213102 (2012); https://doi.org/10.1063/1.4767653
Submitted: 02 October 2012 • Accepted: 01 November 2012 • Published Online: 21 November 2012
S. M. Sadeghi
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Appl. Phys. Lett. 101, 213102 (2012); https://doi.org/10.1063/1.4767653
© 2012 American Institute of Physics.
101, 213102
APPLIED PHYSICS LETTERS 101, 213102 (2012)
Quantum coherence effects in hybrid nanoparticle molecules in the presence
of ultra-short dephasing times
S. M. Sadeghia)
Department of Physics and Nano and Micro Device Center, University of Alabama in Huntsville, Huntsville,
Alabama 35899, USA
(Received 2 October 2012; accepted 1 November 2012; published online 21 November 2012)
We study quantum coherence effects in single nanoparticle systems consisting of a semiconductor
quantum dot and a metallic nanoparticle in the presence of the ultra-short dephasing times of the
quantum dots. The results suggest that coherent exciton-plasmon coupling can sustain the collective
molecular resonances (plasmonic meta-resonances) of these systems at about room temperature. We
investigate quantum optical properties of the quantum dots under this condition, demonstrating
formation of ultranarrow gain and absorption spectral lines. These results are discussed in terms of
plasmonic normalization of coherent population oscillation and the collective states of the
C 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4767653]
nanoparticle systems. V
Quantum coherence in semiconductor quantum dots
(QDs) offers critical steps toward many technological applications, ranging from manipulation of qubits in quantum
logic gates1 to quantum devices,2 nanoswitches,3 etc. Numerous coherent properties common to ideal quantum systems
such as entanglement,4 photon anti-bunching,5 and modification of spontaneous emission rate by a micro-cavity6 have already been demonstrated in various types of QDs. Recent
reports have also shown that QDs can support Mollow spectra
very similar to those in atomic systems.7
One of the most prominent issues facing investigation of
quantum coherence effects in QDs is their ultra-fast polarization dephasing. The rate of this process is not restricted to
real transitions, but rather virtual transitions involving elastic
exciton-phonon scattering can play major roles. In fact,
except for very low temperatures, the time scale of such scattering process (pure dephasing) is very short, reaching several hundreds of femtoseconds at room temperature.8 Our
objective in this paper is to show coherent molecular-like
resonances (plasmonic meta-resonances (PMR)) of nanoparticle systems consisting a QD and a metallic nanoparticle
(Fig. 1(a)) can remain sharp and distinct even when the pure
dephasing times of the QDs are of the same order of magnitudes of those at room temperature. PMR is associated with
two collective states of such hybrid systems formed when
they interact with a laser field, generating quantum coherence in the QDs. Our investigation suggests that coherent
exciton-plasmon coupling caused by quantum coherence in
such hybrid systems forms a self-sustaining mechanism that
allows its impacts survive despite the ultra-short pure
dephasing times of the QDs. We investigate quantum optical
properties of such QDs under these conditions using linear
response theory, demonstrating formation of ultra-narrow
gain and spectral absorption lines. The results are discussed
in terms of the impact of plasmons on coherent population
oscillation (CPO) in the QDs and the collective states of the
QD-metallic nanoparticle (MNP) systems.
a)
Electronic mail: seyed.sadeghi@uah.edu.
0003-6951/2012/101(21)/213102/5/$30.00
The results of this paper are particularly important for
the extensive on-going research investigating the effects of
quantum coherence in QD-MNP systems. These include the
way these effects can be used to induce Rabi frequency,9,10
control speed of light,11 and envision optical nanoswitches
that can be triggered by the change of refractive index of
environment.3 They also include application of coherent
exciton-plasmon coupling to study bistability,13–16 hysteresis,15 and control energy dissipation rates in MNPs and investigate strong coupling between QDs.12,13,17–19 In all of these
investigations, however, the linewidths of the QDs were considered very narrow (few leV), corresponding to nanosecond
scale polarization dephasing time. The results of this paper
will pave way for the investigation of these effects at elevated
temperatures and even at room temperature and open the horizon for richer and more applicable study of coherent phenomena in QDs.
Of particular relevance to this paper is dipole coherence
between the fundamental exciton (j2i) and ground (j1i) states
of a QD. Optically induced coherence between these states
degrades over time by decay of the amplitude of either
states and their coupling to phonons. The latter changes the
relative phase without decay of the individual probability
amplitudes (pure dephasing). Therefore, the total decoherence
FIG. 1. Schematic illustration of the QD-AgNR system studied in this paper
(a) and its electronic system (b). RF and KF refer, respectively, to the FRETinduced energy relaxation and FRET rate to the AgNR.
101, 213102-1
C 2012 American Institute of Physics
V
213102-2
S. M. Sadeghi
rate (c12 ) between j1i and j2i is given by c12 ¼ ðC1 þ C2 Þ=
2 þ cp12 , where Cj refers to the energy relaxation rate of jji
(j ¼ 1 and 2) and cp12 is the pure dephasing rate.
Our system includes a QD and a Ag nanorod (AgNR)
with semimajor as and semiminor bs (Fig. 1(a)). The centerto-center distance between the QD and AgNR is R. As schematically shown in Fig. 1(b), the j1i-j2i (1-2) transition of
the QD with frequency x0 is considered near resonantly
driven by a laser beam with the frequency xl ðEðtÞ ¼ E0 cos
ðxl tÞÞ. We consider strongly confined QD structures wherein
the energy separation between the exciton states is large. In
this limit, as shown in Ref. 20, the diagonal electron-phonon
interactions determine the pure dephasing process and the
Hamiltonian involving the QD and phonons can be diagonalized. Therefore, the ground and excited states of the coupled
exciton-phonon system can be described in terms of eigenfunctions of the uncoupled system.20 Considering this, we
adopt a semiclassical approach wherein the MNP and interaction of the QD-MNP system with the laser beam are
treated classically, while the QD is addressed quantum
mechanically using the density-matrix formalism. We consider quasi-steady state interaction between the QD and the
laser beam, leading to a time-independent phonon-carrier
coupling.21 Considering these, the impact of pure dephasing
is treated in a phenomenological way.20,22
The prominent effects of quantum coherence in QDMNP systems include normalization of the transition energies of the QD and its Forster resonance energy transfer
(FRET) rate to AgNR by the coherent exciton-plasmon coupling. Including these effects, we obtained the equation of
motion for the density matrix of the QD (q) using von Neumann master equation with Lindblad terms23
dq
i
pdep
¼ ½H$ ðqÞ; qðtÞ þ ð£qÞspon
ðqÞ: (1)
$ ðqÞ þ ð£qÞ$
dt
h
Here H$ ðqÞ is the Hamiltonian of the QD wherein $ refers
to the self-normalization process caused by its dependency
on q. H$ ðqÞ is given as
X
hxj rjj þ h½Xr12 ðqÞr21 þ H:C:;
(2)
H$ ðqÞ ¼
j¼1;2
where rij ¼ jiihjj and hxj refers to the energy of jji. Xr12 represents coherently normalized Rabi frequency of the QD,
2
defined as Xr12 ðqÞ ¼ Xef f þ gq21 . Here Xef f ¼ X012 ð1 þ 2cab
R3 Þ
l12 E0
and X012 ¼ 2
hef f are, respectively, referring to the Rabi frequency of the QD in the presence of pure plasmonic effects
(without any coherence effects) and in the absence AgNR
(very large R). ef f ¼ ð20 þ s Þ=30 , where 0 and s are the
dielectric constants of the background and the QD materials,
respectively. l12 is the dipole moment associated with the 12 transition. We approximate AgNR with a spheriod with
polarization c ¼ ½m ðxÞ 0 =½30 þ 3jðm ðxÞ 0 Þ. j is
the depolarization factor of the spheroid when the incident
electric field of the laser is polarized along its semimajor
diameter.19 We describe AgNR with the local dynamic
dielectric function m ðxÞ ¼ IB ðxÞ þ D ðxÞ, which is a combination of the contribution of d electrons ðIB ðxÞÞ and s
electrons ðD ðxÞÞ. g is given by g ¼ 4cl212 as b2s =½
he2ef f R6 . It
Appl. Phys. Lett. 101, 213102 (2012)
has been shown that Im½g is the transition shift caused by
plasmonic field and Re½g is the FRET rate (CF ) from the QD
to AgNR.24
ð£qÞspon
$ ðqÞ in Eq. (1) refers to spontaneous radiative
decay term of the excitons given by19
ð£qÞspon
$ ðqÞ ¼
C2
ð2r12 qr21 r21 r12 q qr21 r12 Þ;
2
(3)
pdep
ðqÞ is the pure dephasing rate term of such
and ð£qÞ$
excitons obtained from
pdep
ðqÞ ¼
ð£qÞ$
cp12
ð2rz qrz rz rz q qrz rz Þ:
4
(4)
In this equation rz ¼ r22 r11 . Considering these, the density matrix equations were obtained as
q_ 11 ¼ 2Im½Xef f q21 þ RF þ C2 q22 C1 q11 ;
(5)
q_ 22 ¼ 2Im½Xef f q21 RF C2 q22 ;
(6)
q_ 21 ¼ ½iðEp hxÞ þ KF þ c12 q21 iXef f d:
(7)
In Eq. (7) Ep ¼ hx0 Re½gd and KF ¼ CF d with d ¼
q11 q22 refer, respectively, to the normalized energy of the
QD (the 1-2 transition) and the contribution of FRET from
the QD to the MNP to the polarization dephasing rate of the
QD (Fig. 1(b)). The fact that both Ep and KF are normalized
by d indicates how the combined effects of quantum coherence and plasmons influence interaction of the laser field
with the QD-MNP system. Additionally, Eqs. (5) and (6)
show that FRET process can act as a non-radiative decay
channel for the excitons in the QD, which is represented by
RF ¼ 2CF jq21 j2 (Fig. 1(b)). This suggests that in the presence of quantum coherence FRET happens efficiently only
when jq21 j2 have noticeable values.25
From Eqs. (5)–(7) we can find the plasmonic field
enhancement factor, defined as the ratio of the field intensity
experienced by the QD in the presence of the MNP (Ief f ) to
that in the absence of it (I0 ). In the presence of quantum coherence this factor is given by Pcoh ðxÞ ¼ jXef f þ gq12 j2 =
jX012 j2 . Considering this equation the effective field intensity
experienced by the QD can be obtained from Ief f ¼ Pcoh
ðxÞI0 .
To obtain the linear absorption of the QD (Aðxp Þ), we
use Aðxp Þ / Re½P1 ðz; t0 Þ P2 ðz; t0 Þjz¼ixp , wherein P1 ðz; t0 Þ
¼ hPþ ðt0 ÞP^ ðz; t0 Þi; P2 ðz; t0 Þ ¼ hP^ ðz; t0 ÞPþ ðt0 Þi, and Pþ
and P are, respectively, the positive and negative frequency
components of the system polarization.26 P^ ðz; t0 Þ is the
0
Laplace transform of P ðt ¼ s þ t Þ with respect to the time
interval s ¼ t t0 ðs > 0Þ.26 xp refers to the frequency of the
probe field. Under steady state (ss) condition ðt; t0 ! 1Þ
Aðxp Þ is obtained as
ss
ss
Aðxp Þ ¼ Re½l212 fqss
21 ½R32 ðzÞ R31 ðzÞ R33 ðzÞ½q11 q22 g:
(8)
Here Rnm are the elements of RðzÞ ¼ ðzI LÞ1 , wherein I
is a 4 4 identity matrix and L contains the coefficients of
qij in Eqs. (5)–(7) and those of q_ 12 . Similar technique was
213102-3
S. M. Sadeghi
Appl. Phys. Lett. 101, 213102 (2012)
also used to calculate the resonance fluorescence (RF) spectrum (Sf lu ¼ Re½P2 z¼ixp ), leading to the following:26,27
Sf lu ðxp Þ ¼ l212 Re½R3;1 ðzÞq21 þ R3;3 ðzÞqss
22 z¼ixp :
0.5
(9)
0.4
ss
ρ22
0.35
5
0.3
0.25
0.2
0.15
−20
21
0.04
20
(a)
I0=34.72
(b)
0.1 I0=33.62
0.05
0.02 1
0
0.1
0.5
−0.05
0
−0.02
0
−20 0 20
−20
−0.1
0
20
2.5 I =34.72
0
2
2.5
3
incoh
10
Therefore, for a given I0 , we can find two laser frequencies
where the D-B transition can happen. Conversely, if we set
xl (or D21 ), this transition occurs when I0 is adjusted such
that the frequency of one of the abrupt changes (edges)
matches xl . In the case of Fig. 2, for example, we set xl
¼ x0 and adjusted I0 to 34 W=cm2 to allow the right
edge happens at xl ¼ x0 or D21 ¼ 0 (thick solid line). When
I0 < 34 W=cm2 this edge shifts to negative D21 , leaving the
system in the D state (dashed-dotted line).
Considering this, to study quantum optical properties of
the QD, in Fig. 4 we study its linear absorption (a), (b) and
RF (c), (d) spectra when I0 ¼ 34:72 W=cm2 (B state) and
33.62 W=cm2 (D state) and cp12 ¼ 2 ps1 . The insets in Figs.
4(a) and 4(c) show the spectra when plasmonic effects are
ignorable (R ¼ 200 nm). The results suggest a dip (hole
Sflu (a.u.)
2
(kW/cm )
2
I
eff
3
0
(meV)
2
FIG. 3. Spectra variation of qss
22 for different I0 (legends in W=cm ) and
cp12 ¼ 2 ps1 . All other specifications are the same as those in Fig. 2.
0
4
−10
Δ
A (a.u)
In the following we consider es ¼ 6 (CdSe-based material), e0 ¼ 4:2 (silicon nitride), as ¼ 6; bs ¼ 4, and R ¼ 9 nm.
Under these conditions the plasmonic peak of AgNR happens
at 2.415 eV. We consider the QD transition has the same
energy (
hx0 ¼ 2:415 eV) and C2 ¼ 0:6 ns1 . Fig. 2 shows
the results for Ief f under steady state and D21 ¼ hðx0 xl Þ ¼ 0 when cp12 ¼ 2 ps1 (circles), 2.5 ps1 (square), and
3 ps1 (triangle) when the laser intensity (I0 ) is varied. These
results suggest a clear switching process, which is smeared
out when cp12 ¼ 3 ps1 . Considering the results for the case
when quantum coherence is ignored (dashed line), it is clear
than even when cp12 ¼ 3 ps1 we can see Ief f shows a nonlinear power-dependency resulted from quantum coherence.
The results in Fig. 2 suggest that even when the QD
dephasing time is less than 500 femtosecond, the QD-AgNR
system supports two states: one with high Ief f (B state) and
the other with much smaller Ief f (D state). These results also
show that the laser intensities where the transition between
these states happens (Ic ) varies with cp12 . For the same system
as in Fig. 2 the spectral variations of qss
22 as a function of D21
for different values of I0 (legends) are shown in Fig. 3. For
any given I0 there is a frequency range where qss
22 0:5 (or
d 0). Therefore, in this range Ep hx0 , and KF and RF
are about zero, suggesting that although the QD is very close
to AgNR, the plasmonic shift and broadening of the 1-2 transition are very small. Under these conditions FRET rate from
the QD to AnNR is suppressed. The fact that in this range
qss
22 0:5 also suggests that the QD feels a strong field and
the QD-AgNR system is in B state. For this reason, the frequency ranges where qss
22 0:5 are called “bright windows,”
since in these ranges the QD emits efficiently. The sharp falls
in Fig. 3 refer to the transition from B to D states.
Fig. 3 shows that the widths of the bright windows,
defined by the two abrupt changes in qss
22 , depend on I0 .
35.04
34.32
34
33.9
33.85
0.45
(c)
−100
1
2
0.8
1.5
0.6
1
1
−0.1
−0.1
0
100
I0=33.62
0.8
0.4
0.1
0
(d)
0.4
−0.1
0
0.1
0.5
1
0.5
0
0
25
30
2
I (W/cm )
35
0
FIG. 2. Variation of the effective field intensity experienced by the QD as
function of I0 for different cp12 (legends in ps1 ). The dashed line obtained in
the absence of quantum coherence.
0
−20
0.2
−20
0
20
0
Δp (meV)
20
0
−100
0
100
Δp (meV)
FIG. 4. Linear absorption and RF spectra of the QD when I0 ¼ 34:72
W=cm2 (a), (c) and 33.61 W=cm2 (b), (d) and cp12 ¼ 2 ps1 . All other specifications are the same as those in Fig. 3. The insets in (a) and (c) show the
spectra when R ¼ 200 nm, and those in (b) and (d) represent the close-looks
of the narrow features in D state and Dp ¼ hðx0 xp ).
213102-4
S. M. Sadeghi
Appl. Phys. Lett. 101, 213102 (2012)
burning) in the absorption peak. With the reduction of R the
dip becomes deeper and eventually slightly negative when
R ¼ 9 nm, i.e., in B state (a). When the system is transferred
to D state the spectrum becomes broadened by more than
three times (b), while the dip becomes much narrower and
significantly negative, generating relatively large gain. The
inset in Fig. 4(b) shows a close look at the dip, suggesting a
Lorentzian-like lineshape. In the case of RF, for R ¼ 200 nm
we see a sharp peak. This peak converts to a hole burning
in B state and again to an ultra-narrow peak in D state
(Fig. 4(d)).
To investigate the physics behind the results shown in
Fig. 4, in Fig. 5(a) we study the way Pcoh changes with time
when cp12 ¼ 2 ps1 and I0 ¼ 34:1 W=cm2 while the system is
interacting with a laser field with the intensity profile as
shown in the left inset of Fig. 5(a). The results suggest a time
delay in Pcoh . This delay increases with reduction of I0 until
it becomes infinite when I0 reaches Ic (the laser intensity
where the B-D transition happens). This shows that in the
time domain when I0 > Ic , for a period of time the system is
in D state, before it switches to B state. Once I0 < Ic the system remains in the D state at all times (not shown). The right
inset in Fig. 5(a) shows that KF is quite efficient during the
time delay (in D state). In B state, however, it becomes relatively very small. This explains the broadenings seen in
Fig. 4. Variations of KF with time also show that the time
delay is caused by the dynamics suppression of excitation of
the QD via normalization of its linewidth (KF ) and effective
energy (Ep ) with exciton population.3 For low field intensities both Ep and KF are significant, suppressing the impact
of the laser field. Therefore, when the laser intensity ramps
up (risetime of the laser reaches the QD-MNP system), the
system resists against excitation, causing the time delay.3
Fig. 5(b) suggests that when cp12 ¼ 3 and I0 ¼ 27:75
W=cm2 , the time delay change is smeared out. The inset
shows that under these conditions KF remains effective at all
times and does not show any step-like changes as seen in
Fig. 5(a). Note that in Fig. 5 we ignored the time dependency
of the exciton-phonon scattering process. This can be justified considering the fact that KF in the case of Fig. 5(b) is
rather overwhelming, reaching 5 ps1 .
It has been shown that in the medium field limit and in
the presence of squeezed vacuum or optical cavities the linear
absorption and RF of 2-level atomic systems can support
60
25
40
B
50
I0
20
20
0
20
20
10
D
10
10
0
0
25
500
20
D
(a)
200
0
0
15
10
B
500
400
600
Time (ns)
5
5
0
0
800 0
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40
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however, were obtained in the low field intensity limit,
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larger than its Rabi frequency (X012 ). Therefore, the system
does not support Mollow spectrum or dressed states. The
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4(c)) can be explained in terms of coherent population oscillation (CPO). This process is known to happen when the population of the ground state oscillates at the beat frequency of
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characteristic features.
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the bright windows, i.e., when the system was in B state. In
D state the exciton population is reduced via large KF (inset
of Fig. 5(a)) and suppression of the effective field of the QD
(Fig. 2). The results in Fig. 4(d), however, suggest a normal
behavior for RF, similar to that in the inset of Fig. 4(c),
although the spectral line becomes much narrower with longer amplitude. The reason behind this can be related to the
fact that while in B state the optical features of the QD are
decided by coherent exciton-plasmon coupling, in D state
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effects as if the laser field nearly does not exist (no coherent
effect), and (ii) coherent generation of a state at the laser
wavelength. The former leads to plasmonic broadening via
large value of KF and assists formation of the ultra-narrow
spectral line in Fig. 4(d). The gain seen in Fig. 4(b), however, is the result of transfer of energy from the applied laser
field to the probe field.
In conclusion, we showed coherent exciton-plasmon
couplings in QD-MNP systems can form a self-sustaining
mechanism that allows them to maintain their impacts despite the very short decoherence times of the QDs. We investigated this by demonstrating how coherent collective
resonances of such systems could survive when these times
were less than 500 femtoseconds. The quantum optical properties of the QDs in such systems showed unique gain and
absorption spectral features.
(b)
200
0
500
400
600
Time (ns)
800
FIG. 5. Variation of the Pcoh as a function of time when cp12 ¼ 2 ps1 (a) and
3 ps1 (b). The left inset in (a) shows the time dependency of the applied
laser field, and the right insets represent the corresponding variation of KF
with time.
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