Slow light in semiconductor quantum dots: effects of non-Markovianity and correlation of dephasing reservoirs

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by  D. Mogilevtsev

Slow light in semiconductor quantum dots: effects of non-Markovianity and
correlation of dephasing reservoirs
D. Mogilevtsev1,2, E. Reyes-G´omez3,4, S. B. Cavalcanti5, and L. E. Oliveira4
1Centro de Ciˆencias Naturais e Humanas, Universidade Federal do ABC, Santo Andr´e, SP, 09210-170 Brazil
2Institute of Physics, NASB, F. Skarina Ave. 68, Minsk, 220072, Belarus
3Instituto de F´ısica, Universidad de Antioquia UdeA, Calle 70 No. 52-21, Medell´ın, Colombia
4Instituto de F´ısica, Universidade Estadual de Campinas - Unicamp, Campinas - SP, 13083-859, Brazil
5Instituto de F´ısica, Universidade Federal de Alagoas, Macei´o - AL, 57072-970, Brazil
(Dated: January 8, 2016)
A theoretical investigation on slow light propagation based on eletromagnetically induced trans-
parency in a three-level quantum-dot system is performed including non-Markovian effects and
6 correlated dephasing reservoirs. It is demonstraonated that the non-Markovian nature of the pro-
1 cessisquiteessentialevenforconventionaldephasingtypicalofquantumdotsleadingtosignificant
0 enhancement or inhibition of the group velocity slow-down factor as well as to the shifting and
2
distortion of the transmission window. Furthermore, the correlation between dephasing reservoirs

n may also either enhanceor inhibit non-Markovian effects.
a
J PACSnumbers: 42.50.Gy, 42.50.Nn,42.50.Ar,and78.67.Hc

7

] I. INTRODUCTION as suggestedin experimental22 andtheoretical23 studies.
h Reservoir correlations do not decay quickly enough and
p
thedensityofreservoirstateschangessignificantlyonthe
- Slow light group velocity propagation has revealed it-
nt self to be a key stone in the construction and design scale of reservoir-system interaction constants and Rabi
frequencies of driving fields. It is important to note that
a of variable delay lines which are of great importance in
non-Markovian dephasing in quantum dots is responsi-
u thesynchronizationofopticalsignalsandsignalbuffering
q in all-optical communication systems. One of the most ble for phenomena such as, for instance, the damping of
[ Rabi oscillations and excitation-induced dephasing,24–30
promising approaches is the one implementing electro-
1 magnetically induced transparency (EIT).1 It has been phonon-induced spectral asymmetry,30–33 and interfer-
ence between phononic and photonic reservoirs.34–36
v experimentally demonstrated that in EIT schemes it is
0 possible to obtain a slow-down factor of 107 in gases, Tothe bestofourknowledge,onlyMarkovianandun-
4 such as Rb vapor2 and cold cloud of sodium atoms,3 correlated dephasing have been considered in semicon-
5
or solid-state systems as Pr doped4 Y SiO . However, ductor EIT systems. In the present study, we show,
1 2 5
the transmission bandwidths obtained in these systems for semiconductor quantum-dot systems, that the non-
0
. are too narrow5 (about 50-150 KHz) for optical buffers. Markovian nature of dephasing leads to a significant
1 Semiconductor structures such as quantum wells6,7 and modification of the group velocity slow-down factor in
0
quantum dots8–12 may offer much broader transmission comparison with the Markovian one. Moreover, it is
6
bandwidths (about a few GHz) at the cost of a smaller demonstrated that the absorption spectrum becomes
1
: slow-downfactorandwithEITbuffersoperatingatroom asymmetric and the transmission window is shifted and
v temperature.13,14 To open up possibilities in light con- modified. It is interesting that, in contrast with the
i
X trol, one may combine EIT semiconductor structures dampingofRabioscillations,24–27 thechangesundergone
r withothersystemscapableofslowinglight,i.e.,photonic by the slow-down factor and the transmission window
a crystals15–17 orcoupledquantum-dotheterostructures.18 are actually first-order effects on the frequency of the
driving field associated with the Rabi oscillations. They
One feature of the semiconductor nanostructures such
as quantum dots and wells is the interaction with the may occur even in situations where the driving-induced
dephasing and the damping of driven Rabi oscillations
substratehost. Forexample,quantumdotsinteractwith
phonons or with carriers captured in traps in the vicin- do not take place. Furthermore, correlations between
dephasing reservoirs is also considered. Correlation be-
ity of the quantum dot. The influence of the surround-
ings leads to dephasing and to energy loss of the semi- tween losses were already shown to lead to a number of
non-trivial effects in the dynamics of open systems.37–40
conductor nanostructure. The mostcommondephasings
In this respect, well-known decoherence-free subspaces
andenergylossesmaybedescribedbyMarkovianmaster
result from different couplings of the system with the
equationsusuallywrittendowninthe so-calledLindblad
form19 (see, for example, the recent works by Colas et same reservoir, and may be used for avoiding decoher-
al20 and Marques et al21). Markovianity arises when enceofthequantumstates.37,38 Correlatedlossesmaybe
exploited to create nonlinear loss in deterministic non-
correlationsofdephasingordissipativereservoirsrapidly
classical states generation39 and to produce excitation
decay on the typical time-scale of the nanostructure dy-
flow like heat while retaining coherence.40
namics. However, it is well known that the dephasing
processinsolidsisquiteoftenofanon-Markoviannature The outline of the present study is as follows. In Sec-
2
InAs quantum-dot system, dephasing times were about
several tens of picoseconds at temperatures of 50 K - 80
K, whereas energy loss occurred on the scale of several
nanoseconds). In the rotating-wave approximation and
interaction representation, the three-level quantum dot
scheme depicted in Fig. 1 may be described by the fol-
lowing generic Hamiltonian
V(t) = ~[∆ +∆ +R (t)]σ +~[∆ +R (t)]σ
s p 3 33 s 2 22
Ω Ω
+ ~ p(σ +σ )+~ s(σ +σ ), (1)
32 23 21 12
2 2
FIG.1: Pictorialviewofathree-levelquantumdotdrivenby
a strong pump field with frequency ωp and Rabi frequency where, for simplicity, we assume Ωp and Ωs to be real,
Ωp. R2 and R3 denote dephasing reservoirs coupled to the operators σkl =|kihl| (k,l =1, 2, 3), |ki describes the k-
corresponding levels described by states |2i and |3i, respec- th state of the system, and detunings are ∆ =ω −ω
s 21 s
tively. Thesignalfieldwithfrequencyωs andRabifrequency and∆ =ω −ω . TheHermitianoperatorR =R (t)
p 32 p k k
Ωs. Dipole allowed transition frequencies are denoted by ω32 describes the dephasing reservoirsinfluencing the transi-
and ω .
21 tion to the k-th state. We notice that one may exclude
an action of a dephasing reservoir on the lower state by
3
tion II we introduce the model of the three-level quan- using σ =1. Thus,eachoperatorR containsvari-
kk k
tum dot in a ladder configuration under the action of k=1
both a strong pump and of a weak signal optical fields. ables oPf two reservoirs (see, for example, the study by
The master equation and corresponding stationary solu- Kaer et al.32). Here we do not specify the exact nature
tions are presented in section III, whereas the suscepti- of the dephasing reservoirs. It might include all types
bility, absorption and slow-downfactor are given in Sec- of pure dephasing encountered in quantum dots beyond
tion IV. In Section V, the influence of non-Markovian the independent linear boson model commonly used to
effects onthe slow-downfactorandabsorptionaswellas describe the phonon interaction with the dot, such as
some examples are analyzed. Also, a comparison with effects of possible quadratic coupling of phonons to the
the Markovian case is demonstrated. Finally, discussion dot41 or phonon-phonon scattering.42 We require only
and conclusions are presented in Sections VI and VII, the existence of the quantities
respectively.
t
D±(δ)= lim dτK±(t,τ)eiδτ (2)
kl t→∞ kl
Z
II. THE THREE-LEVEL QUANTUM-DOT 0
MODEL
for any real δ, where K+(t,τ) = hR (t)R (τ)i and
kl k l
K−(t,τ) = hR (τ)R (t)i. For realistic many-system
TheexperimentalsetupneededtoobserveEITinvolves kl k l
reservoirs one usually has hR (x)R (y)i → 0 for |x −
twohighlycoherentopticalfieldsinteractingwithathree k l
y|, x, y →+∞ and the quantities (2) do exist, defining,
level system in various schemes. Here, we are interested
for example, asymptotic decay rates obtained with the
in a quantum dot model and therefore we choose a lad-
time-convolutioness master equation.43
der scheme for EIT as depicted in Fig. 1: a weak signal
field is tuned near resonance with the |1i → |2i transi-
tion, while a strong pump quasi-classical field is tuned
III. THE MASTER EQUATION
with the |2i → |3i transition. We denote by ω and ω
s p
the optical frequencies of the signal and pump fields, re-
Thereisastrongpumpfielddrivingtransitionbetween
spectively, and Ω , Ω the Rabi frequencies associated
s p
states |2i and |3i of the three-level quantum dot. To
with the signal and pump fields. Such a scheme was
considered by Kim et al.9 for a strained GaAs-InGaAs- derive a master equation accounting for effects of the
strong dot-field interaction, it is necessary to transform
InAs quantum-dot system. We assume that ω and ω
21 32
to a dressed picture with respect to the operator
are sufficiently different so that the driving by the ω
p
strong pump field (with Ω Rabi frequency) only occurs
p Ω
between upper states |2i and |3i. Similarly, the ω weak V =~∆ σ +~ p(σ +σ ). (3)
s d p 33 32 23
2
signal field (with Ω Rabi frequency, |Ω | ≪ |Ω |) acts
s s p
only between states |1i and |2i. Also, to illustrate the The dressed Hamiltonian is
effects of the non-Markovian character of the dephasing
process, we assume that energy loss occurs on times far V(t) = ~[∆ +R (t)]S (t)+~[∆ +R (t)]S (t)
s 3 33 s 2 22
exceedingthetypicaldephasingtimeofthequantumdot Ω
(e.g., in the study by Kim et al.,9 for a GaAs-InGaAs- + ~ s[S21(t)+S12(t)], (4)
2
3
where the dressed operators are S (t) = U†(t)σ U(t), Notice that the form of the equations for coherences
kl kl
and U(t) = exp(−iV t/~). Thus, for the population [Eqs. (8a) and (8b)] is the same for both Markovian
d
operators, one has24 and non-Markovian cases. However, a non-Markovian
process leads to time-dependent coefficients in Eqs. (8)
S33(t) = S33(t)σ33+S23(t)σ23+S32(t)σ32 and also to their dependence on the driving field. The
+ S22(t)σ22 (5a) time-dependent decay rates in Eqs. (8) are γk(t) =
γM(t) + γC(t), where the γM(t) represent decay rates
k k k
and for zero pump,
S22(t)=σ33+σ22−S33(t), (5b) t
γM(t)= dτK−(t,τ), k =2, 3, (9)
where k kk
Z
0
1
S (t)= 1+c2+s2cos(Ω t) , (6a)
33 2 R and the driving-dependent parts of the total decay rates
are
(cid:2) (cid:3)
s2 t
S (t)= [1−cos(Ω t)], (6b)
22 2 R γ2C(t)= dτS22(τ −t) K3−2(t,τ)−K2−2(t,τ) (10a)
Z
and 0 (cid:2) (cid:3)
s and
S (t)= {c[1−cos(Ω t)]+isin(Ω t)}, (6c)
23 R R
2
t
with Ω2R = ∆2p + Ω2p, c = ∆p/ΩR, s = Ωp/ΩR and γ3C(t)= dτ [1−S33(τ −t)] K2−3(t,τ)−K3−3(t,τ) .
S32(t)=S2∗3(t). Z0 (cid:2) (cid:3)
Notice that here we assume low temperatures, so it (10b)
is not necessary to consider multi-phonon processes and Thetime-dependentcross-coherencecouplingparameters
to implement polaron transformations to account for in Eqs. (8) are
them. For low temperatures one may use the time-
convolutioness master equation to describe the dynam- t
ics of the dot dressed by the driving field32 and derive ν (t)= dτS (τ −t) K−(t,τ)−K−(t,τ) (11a)
2 32 32 22
in a standard manner the master equation for the ρ¯(t)
Z
dressed density matrix of the dot.43 Up to second order 0 (cid:2) (cid:3)
onthe interactioninvolvingreservoirs,onearrivesatthe and
following equation for the dressed density matrix aver-
t
aged over the reservoirs
ν (t)= dτS (τ −t) K−(t,τ)−K−(t,τ) . (11b)
3 23 33 23
dρ¯(t) = −i[∆s{S22(t)+S33(t)},ρ¯(t)] Z0 (cid:2) (cid:3)
dt
+ i[Ω {S (t)+S (t)},ρ¯(t)] Notice that the decay rates introduced by Eqs. (9) and
s 21 12
(10) might include imaginarycomponents corresponding
t
to driving-dependent frequency shifts. Eqs. (9), (10),
− dτh[V (t),[V (τ),ρ¯(t)]]i, (7)
R R and(11)giveaclearideaabouttheinfluenceofthereser-
Z0 voir correlation on the dynamics. Complete correlation
between reservoirs, i.e., K−(t,τ) = K+(t,τ), k = 2, 3,
where V (t) = R (t)S (t)+R (t)S (t). Returning to kl kl
R 3 33 2 22 washes out the influence of an arbitrarily strong pump
thebarebasis,oneobtainsthefollowingequationsforthe
driving on the coherences dynamics. Onthe other hand,
off-diagonal (coherence) elements of the density matrix
complete anti-correlationenhancesthe driving influence.
ρ =hk|ρ|li,
kl
d
ρ13(t) = −[γ3(t)−i(∆s+∆p)]ρ13(t) IV. SUSCEPTIBILITY, ABSORPTION AND
dt
SLOW-DOWN FACTOR
i i
+ ν (t)− Ω ρ (t)+ Ω ρ (t), (8a)
3 p 12 s 23
2 2
(cid:20) (cid:21) Now let us consider the solution of the system of Eqs.
and (8)inthet→∞limit. Weassumethat lim ρkl(t)≡ρsktl,
t→∞
lim ν (t) ≡ ν , and introduce real asymptotic values of
d i t→∞ k k
ρ (t) = −[γ (t)−i∆ ]ρ (t)+ ν (t)− Ω ρ (t)
12 2 s 12 2 p 13 the dephasing rates
dt 2
(cid:20) (cid:21)
i
+ Ω [ρ (t)−ρ (t)]. (8b) γ =Re lim γ (t) , (12)
s 22 11 k k
2 t→∞
h i
4
(a) (a)
3
0.8 T=5 K
T=15 K
2 2
γe, 0.6 T=45 K
at
ay r 0.4 ’χy, 1
Dec bilit
0.2 pti
ce 0
s
u
S
0
0 0.5 1 1.5 2 2.5
−1
(b)
1.5 −2
T=5 K
−4 −2 0 2 4
ν2 T=15 K
ant, 1 T=45 K (b)
st
n
o
g c 18
n
pli 0.5 16 1
u
o
C
14
00 0.5 1 1.5 2 2.5 12
Rabi frequency, Ω ’’χ 0.5
p n, 10
o
pti
FIG. 2: (a) γ2 decay rate and (b) ν2 coupling constant as sor 8
vfuonirctcioornrselaotfiothnefuRnacbtiionfregqivueenncbyyΩEpq,.a(c2c6o)r.dAinllgqtuoanthtietieressaerre- Ab 6 −04 −2 0 2 4
definedinunitsofthetypicalRabifrequencyΩ =6.6µeV.9
0 4
Solid, dash-dotted and dotted lines correspond to tempera-
tures T =5 K, 15 K, and 45 K,respectively. 2
0
−3 −2 −1 0 1 2 3
and modified detunings Detuning, δs
δ =∆ +Im lim γ (t) (13a) FIG. 3: (a) Normalized linear susceptibility and (b) absorp-
s s 2
t→∞ tionprofileintheweaknon-Markoviancaseassumingγ2 and
h i γ3 independent of Ωp and ν2 ≈ f2Ωp [cf. Eq. (29) with
and g = g = 0]. Here we take γ = 2γ = 2Ω , ν = 2ν ,
2 3 3 2 0 3 2
δp = 0, and Ωp = 3Ω0. All parameters are given in units
δp =∆p+Im lim γ3(t) . (13b) of Ω0 = 6.6 µeV. Solid, dashed and dash-dotted lines corre-
t→∞ spond to f = 0 Markovian case, f = 0.1 Ω , and f = 0.2
2 2 0 2
h i
Ω , respectively. Other parameters are the same as in the
0
Thestationarysolutionsforthecoherencearethengiven studybyKimet al.9: εbac =13,|µ21|/e=2.1nm,Γ=0.006,
by V =8.9064×102 nm3,andthewavelengthof thesignal field
is 1.36 µm. Inset in panel (b) shows the absorption profiles
(2ν −iΩ )ρst −iΩ ρst in more detail.
ρst = 3 p 12 s 23 (14a)
13 2[γ −i(δ +δ )]
3 s p
and
rate45
(2ν −iΩ )ρst −iΩ (ρst −ρst)
ρst = 2 p 13 s 22 11 . (14b)
12 2(γ2−iδs) ′ Φ ν2+ν3
χ ≈ ε + γ γ δ +Ω
bac ~Ξ 3 3 s p 2
Taking into account that the pump driving field is much (cid:18) (cid:19)
more intense than the signal field (|Ωp| ≫ |Ωs|), one + Φ (δ +δ ) ν ν − Ω2p +(δ +δ )δ (15a)
obtains9fortheχ′ linearsusceptibilityandχ′′ absorption ~Ξ p s 2 3 4 s p s!
5
(a)
105
3
2
ow−down factor110034 ’χuseptibility, −011
Sl S
−2
102 −3
100 101 −4 −3 −2 −1 0 1 2 3
Rabi frequency, Ω
p (b)
20
FIG. 4: Slow-down factor [cf. Eq.(16)] in the weak non- 1
Markovian case assuming ν2 ≈ f2Ωp [g2 = g3 = 0 in Eq.
(29)] and δp = δs = 0. Solid, dashed and dash-dotted lines ’’χ 15
correspond to the f2 = 0 Markovian case, f2 = 0.1 Ω0, and n, 0.5
f = 0.2 Ω , respectively. The other parameters are chosen o
a2s in Fig. 30. orpti10
s
Ab 0
5 −2 0 2
and
0
χ′′ ≈ ~ΦΞγ3 γ3γ2+ Ω42p + ~ΦΞ γ2(δs+δp)2−γ3ν2ν3 , −3 −2 −1 D0etuning,1 δs 2 3 4
!
(cid:2) (15b(cid:3)) FIG. 5: (a) Normalized linear susceptibility and (b) absorp-
whereΞ=[γ γ +Ω2/4−ν ν −(δ +δ )δ ]2+[γ (δ +δ )+ tion profile in the weak non-Markovian case assuming ν ≈
3 2 p 2 3 s p s 2 s p 2
γ3δs+Ωp(ν2+ν3)/2]2, Φ=Γ|µ12|2(ρs1t1−ρs2t2)/ǫ0Θ, εbac f2Ωp [g2 = g3 = 0 in Eq. (29)]. Solid, dashed, dash-dotted
anddottedlinescorrespondtothe(f ,f )=(0,0)Markovian
is the background dielectric constant, µ is the dipole 2 3
12
case,(f ,f )=(−0.2,−0.4)Ω ,(f ,f )=(−0.2,0.4)Ω ,and
moment of the transition between states |1i and |2i, ǫ 2 3 0 2 3 0
0 (f ,f ) = (0.2,−0.4)Ω , respectively. Other parameters are
is the vacuum electric permittivity, and Θ is the volume 2 3 0
chosen as in Fig. 3. Inset in panel (b) shows the absorption
of a single quantum dot. Γ is the optical confinement
profiles in more detail.
factor defined as the fraction of the field intensity con-
fined to the dots.44 Here we assume that our structure
is typical for vertical-cavity quantum dot lasers, where V. NON-MARKOVIAN EFFECTS
lightpropagatesperpendicularto the activelayer. Thus,
Let us consider the asymptotic values of γ and ν
k k
dzI(z)
to establish the non-Markoviandynamics of the system.
Γ≈ζ dRot dzI(z), One then obtains
structure s2 s2
R γ = D−(0)+ [D−(0)−D−(0)]− D−(Ω )
2 22 2 32 22 4 32 R
I(z) being the field intensity as a function of the coor-
dinate along the direction of propagation and ζ is the − D2−2(ΩR)+ D3−2(−ΩR)−D2−2(−ΩR)(cid:2) (17)
ratio of the dots area to the area of the structure in
and (cid:3)
the direction perpendicular to the signal-field propaga-
tion. Roughly, Γ is proportional to the ratio of the total s2 s2
γ = D−(0)+ [D−(0)−D−(0)]− D−(Ω )
dot volume to the volume of the structure. Here we as- 3 33 2 23 33 4 23 R
sume thatthe dotremainsonthe lower|1istate,sothat − D−(Ω )+D−(−Ω )−D−(−Ω )(cid:2). (18)
ρst ≪ ρst ≈ 1. The slow-down factor, defined as the 33 R 23 R 33 R
22 11
ratio of the group velocities of light outside and inside The t → ∞ limit is equivalent to the Ma(cid:3)rkovian ap-
the slowing medium, is given in our case by proximation. We note that, due to the driving field,
γ and ν are dependent on the values of the Fourier-
k k
dn
transforms [cf. Eq. (2)] of the reservoircorrelationfunc-
Υ=n+ω , (16)
s
dωs tions at δ = 0, ±ΩR (hence the name local Markovian
approximation46),whereasforthe traditionalMarkovian
where n= χ′ +iχ′′ is the complex refractive index. approximationthe ratesareonlydependentonthe value
p
6
of the Fourier-transforms at δ = 0. We assume weak and
non-Markovian effects, i.e., if D−(ω) is smooth in the
kl
vicinity of ω = 0 and varies only slightly on the scale
defined by the modified Rabi frequency Ω , then it may
R Ω d
be represented as a polynomial up to second order in ω. ν3 ≈ 2pdω D3−3(ω)−D2−3(ω)
Therefore, Eq. (17) reduces to (cid:12)ω→0
+ Ωp∆p (cid:2)d2 D−(ω)−D−(cid:3)((cid:12)(cid:12)(cid:12)ω) . (24)
Ω2 d2 2 dω2 33 23
γ ≈ D−(0)− p D−(ω)−D−(ω) . (19) (cid:12)ω→0
2 22 4 dω2 32 22 (cid:12) (cid:2) (cid:3)(cid:12)(cid:12)
(cid:12)ω→0 (cid:12)
(cid:2) (cid:3)(cid:12)
(cid:12)
Similarly, one obtains for the γ3 decay rate (cid:12) One may note that, for the calculation of the coherence
coupling parameters,it is sufficient for the transforms of
Ω2 d2 the D−(ω) bath correlation functions to be linearly de-
γ ≈ D−(0)− p D−(ω)−D−(ω) . (20) kl
3 33 4 dω2 23 33 (cid:12) pendent on the frequency to have the non-Markovianity
(cid:2) (cid:3)(cid:12)(cid:12)ω→0 affecting the susceptibility and absorption. As we shall
(cid:12) seebelow,evenfor|D−(Ω )−D−(0)|≪|Ω |, theinflu-
Therefore, the decay rates depend on the sq(cid:12)uared Rabi kl R kl R
ence of non-Markovianity on the susceptibility, absorp-
frequency associated with the driving field. Here we
tionandthe slow-downfactormay be quite pronounced.
note that such a dependence leads to the damping of
the drivenRabioscillations.24 Similarly, forthe coupling To demonstrate that it is quite common that non-
Markovianity leads to significant coherence coupling pa-
parameters from Eq. (2) one obtains
rameters,let us consider a model of pure dephasing pro-
ν = sc D−(0)−D−(0) − s(1+c) D−(Ω ) duced by the reservoir of acoustic phonons at low but
2 2 32 22 4 32 R finite temperature. Such a model has been extensively
− D−(cid:2)(Ω ) + s(1−c(cid:3)) D−(−Ω )(cid:2) used for describing pure dephasing.25–35 In the interac-
22 R 4 32 R tion picture with respect to the phonon reservoir, the
− D−(−Ω(cid:3)) (cid:2) (21) reservoir operators are described by43
22 R
and (cid:3)
sc s
ν = [D−(0)−D−(0)]+ (1−c) D−(Ω ) R (t)= g b e−iwlt+b†eiwlt , (25)
3 2 33 23 4 33 R k kl l l
− D−(Ω ) − s(1+c) D−(−Ω (cid:2)) Xl (cid:16) (cid:17)
23 R 4 33 R
− D−(−Ω(cid:3)) . (cid:2) (22)
32 R
where the b and b† are annihilation and creation op-
As before, it is straig(cid:3)htforward to show that Eqs. (21) erators of thle phonoln mode with frequency w and the
l
and (22) lead to g are interaction constants (for simplicity, assumed as
lk
real).
Ω d
ν ≈ p D−(ω)−D−(ω)
2 2 dω 22 32 Asanexample,letustakejustonecorrelationfunction
(cid:12)ω→0
+ Ωp∆p (cid:2)d2 D−(ω)−D−(cid:3)((cid:12)(cid:12)(cid:12)ω) (23) [rie.ese.,rvKo2ir2s(.τ,Fto)r] athsseurmesinergvothireraetitsemnopecroartruerlaetTio,n between
2 dω2 22 32
(cid:12)ω→0
(cid:2) (cid:3)(cid:12)
(cid:12)
(cid:12)
∞
K22(τ,t)=Xg22lh[nT(wl)+1]e−iwl(τ−t)+nT(wl)eiwl(τ−t)i=Z dwJ(w)h[nT(w)+1]e−iwl(τ−t)+nT(w)eiwl(τ−t)i, (26)
l 0
where the average number of thermal phonons in the density of states
mode is
w2
~w J(w)=αw3exp −2w2 , (28)
nT(w)=coth (27) (cid:20) c(cid:21)
2k T
(cid:18) B (cid:19)
where α accounts for the interaction strength and w is
c
and the function J(w) is the density of states of the the cut-off frequency. Such a function describes a reser-
phonon reservoir. Let us consider a typical super-Ohmic voirofacousticphononsidentified,forexample,asama-
7
VI. DISCUSSION AND EXAMPLES
Here we wish to investigate the influence of non-
104
or Markovian effects on the susceptibility and absorption
ct
a of a three-level quantum dot. To this end, we compare
wn f the Markovian and non-Markovian regimes and analyze
do103 theinfluenceofnon-Markovianeffectsandcorrelatedde-
−
w phasing reservoirs.
o
Sl IntheMarkovianlimit wechoosethe valuesofparam-
102 eters used by Kim et al.9 in the case of a cylindrical
strained GaAs-InGaAs-InAs quantum-dot system, i.e.,
100 Rabi frequency, Ω 101 εbac =13,|µ21|/e=2.1nm,Γ=0.006,V =8.9064×102
p nm3, and the wavelength of the signal field is 1.36 µm,
whichismuchshorterthanthewavelengthofthedriving
FIG. 6: Slow-down factor [cf. Eq.(16)] in the weak non-
field(12.8µm). Also,itwasassumedthatthe dephasing
Markovian case assuming ν2 ≈ f2Ωp [cf. Eq. (29) with
rate for level 3 was two times larger than for level 2. In
g2 = g3 = 0] and δp = δs = 0. Solid, dashed, dash-dotted
anddottedlinescorrespondtothe(f ,f )=(0,0)Markovian the results shown below we also make this assumption
2 3
case,(f ,f )=(−0.2,−0.4)Ω ,(f ,f )=(−0.2,0.4)Ω ,and for both the dephasing rates and coupling parameters
2 3 0 2 3 0
(f ,f ) = (0.2,−0.4)Ω , respectively. Other parameters are γ = 2γ and |ν | = 2|ν |. Moreover, here we assume
2 3 0 3 2 3 2
chosen as in Fig. 3. that decay rates and coupling constants depend on the
Rabifrequencyassociatedwiththedrivingfield[cf. Eqs.
(19)-(20) and (23)-(24)] as
105 γ ≈γ(0)+ gkΩ2, ν ≈f Ω +g ∆ Ω , (29)
k k 2 p k k p k p p
for k = 2, 3. As mentioned, D− has been repre-
ctor104 sented as a polynomial up to seconkkd order in ω, i.e.,
a
wn f Dk−k(ω) ≈ γk(0) + 2fkω + gkω2 for k = 2, 3. In addi-
o tion, D−(ω) = D−(ω) = 0. It should be noted here
w−d103 that γ(03)2rates mig23ht also include a contribution from
o k
Sl Markovian reservoirs describing energy loss and other
sources of Markovian dephasing. Let us first consider
102
the case ofanegligible changeofthe decayratewith the
100 Rabi frequency, Ωp 101 Rnoanb-iMfraerqkuoevniacny,eiff.ee.c,tγsk((0|)f≫Ω||g2k≪Ω2pγ|(0fo),rkk==22,,33).oWnetahke
k p k
susceptibility, absorptionandslow-downfactor,are then
FIG. 7: Slow-down factor in the weak non-Markovian case. illustrated in Figs. 3 and 4. Fig. 3(a) depicts the sus-
Here [cf. Eq. (29)] f3 = 2f2, g3 = 2g2, δp = δs = 0. ceptibility for the Markovian resonant (f = 0, ∆ = 0)
2 p
Solid,dashed,dash-dottedanddottedlinescorrespondtothe
and two non-Markovian cases (f = 0.1Ω , f = 0.2Ω ,
(f ,g ) = (0,0) Markovian case, (f ,g ) = (0.2,0.0001)Ω , 2 0 2 0
2 2 2 2 0 ∆ =0)asafunctionofthesignal-fieldδ detuning. One
(f ,g ) = (0.2,0.1)Ω , and (f ,g ) = (0.2,0.15)Ω , respec- p s
2 2 0 2 2 0
noticesthattheslopeofthenon-Markoviansusceptibility
tively. Otherparameters are chosen as in Fig. 3.
curve is indeed rather pronouncedly enhanced. In addi-
tion, as indicated by results from the absorption profiles
in Fig. 3(b), non-Markovianeffects shift and narrow the
transmissionbandwidths (the latterisnaturaltoexpect;
jor source of pure dephasing in InGaAs/GaAs quantum see, for example, Tucker et al.5). Moreover, the increase
dots.26Wetakeα=0.4π2ps−2(see,forexample,Hughes in the slow-down factor may be quite large (cf. Fig. 4)
etal.33)andassumeacut-offfrequencyw =1meV.Fig. whereas the transmission bandwidth is not significantly
c
2 displays the dependence of the γ driving-dependent reduced. Non-Markovian effects are phase-sensitive and
2
decay rate [see Eq. (17)] and ν coupling constant [cf. maybe eitherharmfuloradvantageousforslowingdown
2
Eq. (21)] on the Ω Rabi frequency. We have scaled all light. Simultaneouschangesofsignofcouplingconstants
p
the quantities with the typical value9 of the Rabi fre- ν and ν shift the peaks of both the susceptibility and
2 3
quency of the driving field, Ω = 6.6 µeV. One may see absorption to the region of negative δ detunings (see
0 s
that, for low temperatures (5 K-45 K was used for sim- Figs. 5 and 6). However, the change is not completely
ulations in Fig. 2), the dependence of decoherence rates symmetrical and gain in the slow-down factor is smaller
andcouplingconstantsonthedrivingfieldisindeedquite thaninthecaseofν andν bothpositive. Itisinterest-
2 3
pronounced and, quite definitely, may not be ignored. ing to point out that taking ν and ν of opposite signs,
2 3
8
one maysimultaneouslyobtainboththe enhancementof ues of the d deformation potential. If the deformation
k
theslow-downfactorandwideningoftransmissionband- potentials are close in values for different excited states,
width [see inset in Fig. 5(b) and Fig. 6]. themodeldefinedinEq. (25)predictsobliterationofthe
Up to now we have considered decay rates indepen- non-Markovian effect by correlated dephasing. Notice
dent of the Rabi frequency Ω . As we have shown be- that, for an adequate treatment of correlations between
p
fore, such dependencies may be considered through Eq. reservoirs, one needs to appropriately account for phase
(29) by taken g 6= 0 (k = 2, 3). In this case, the non- differences between constants g to develop a realistic
k kl
Markovian nature of the EIT process may destroy the microscopic dephasing model.
efficiency in obtaining slow light. To see this, we turn VII. CONCLUSIONS
to Fig. 7 where the behavior of the slow-down factor is
plotted as a function of the Rabi frequency associated
In conclusion, we have shown that it is necessary to
with the pump field. If decay rates depend on the Rabi
take into account non-Markovian effects when consider-
frequency Ω , one may obtain EIT systems with worse
p ing slowing down light in EIT schemes based on quan-
performance than in the Markovian case. As such de-
tum dots. Non-Markovian behavior is typical for such
pendence becomesmoreremarkable,i.e, asg andg are
2 3 systems, especially in low temperature conditions. We
increased [cf. Eq. (29)], a shifting and narrowing of the
have demonstrated that, for decay rates independent of
transmission window is more noticeable, and a complete
the Rabi frequency associated with the pump field, even
loss of the non-Markovian slow-down effect may be ob-
relativelyweaknon-Markovianeffectsmayleadtosignif-
tained as the Rabi frequency Ω is increased.
p icantenhancementoftheslow-downfactortogetherwith
We note that correlations between reservoirs may af-
thesimultaneousbroadeningofthetransmissionwindow.
fecttheinfluenceofmemoryeffectsontheslow-downfac-
However, if decay rates are considered to be dependent
tor. In a quantum dot system, the interaction between
of the Rabi frequency Ω , a shifting and narrowing of
thedotelectronsanditssurroundingmedium,i.e.,impu- p
thetransmissionwindowmaybeobtained. Furthermore,
rities, phonons, etc., may lead to significant correlations
non-Markovian effects may lead to significant driving-
betweenreservoirs. Moreover,byconstruction,operators
induced dephasing which may inhibit the slowing-down
R (t) (k = 2, 3) contain variables corresponding to the
k effect. Moreover, it is suggested that the presence of
samereservoiractingonthelowerlevelofthedot. Tosee
correlation between reservoirs may remove the harmful
how the correlationbetween reservoirsis inevitably aris-
effects produced by the non-Markovian nature of EIT.
ing in conventional models of dephasing, let us consider
Finally, considering the importance of investigations to
an example of the electron-phonon interaction constants
produceefficientslowlightpropagationinsolid-statesys-
[cf. Eq. (25)] for the present three-level dot. They may
tems, we do hope this work would stimulate future ex-
be represented in the following general form31
perimental and theoretical studies on this subject.
g =C ~|L~| d3~r d |φ (~r)|2−d |φ (~r)|2 eiL~·~r,
kl k k 1 1
q Z
(cid:2) (cid:3) (30)
where L~ is the wave-vector of the l-th phonon mode, d Acknowledgments
k
and φ (~r) are the deformation potential and the elec-
k
tron wave-function of the k-th state of the dot, respec- This work was partially supported by the National
tively,andC isaconstant. Sinceelectronwave-functions Academy of Sciences of Belarus through the program
are localized in the vicinity of the dot, and taking into “Convergence”,andbyFAPESPgrant2014/21188-0(D.
account the small energies of acoustic phonons (in the M.). The authors would also like to thank the Scientific
presentcaseitwouldbemuchlessthan1meV),theden- Colombian Agencies Estrategia de Sostenibilidad (2015-
sities of states of the phononic reservoirs corresponding 2016) and CODI - University of Antioquia, and Brazil-
to states |2i and |3i should be similar. The difference ian Agencies CNPq, FAPESP (Proc. 2012/51691-0)and
in the integrand of Eq. (30) involves essentially the val- FAEPEX-UNICAMP for partial financial support.
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