External-field effect on quantum features of radiation emitted by a quantum well in a microcavity
ABSTRACT We consider a semiconductor quantum well in a microcavity driven by coherent light and coupled to a squeezed vacuum reservoir. By systematically solving the pertinent quantum Langevin equations in the strong-coupling and low-excitation regimes, we study the effect of exciton-photon detuning, external coherent light, and the squeezed vacuum reservoir on vacuum Rabi splitting and on quantum statistical properties of the light emitted by
the quantum well. We show that the exciton-photon detuning leads to a shift in polariton resonance frequencies and a decrease in fluorescence intensity. We also show that the fluorescent light exhibits quadrature squeezing, which predominately depends on the exciton-photon detuning and the degree of the squeezing of the input field.
- [Show abstract] [Hide abstract]
ABSTRACT: We present a study of semiconductor cavity QED effects with squeezed light. We investigate the effects of external squeezed light produced by a subthreshold optical parametric down conversion on the quantum features of the cavity as well as output radiation in the presence of exciton-exciton scattering. It turns out that the width of the spectrum of the cavity field strongly depends on the degree of squeezing. This effect is observed both in weak-and strong-coupling regimes. Moreover, we show that the external squeezed light has a profound effect on the amount of squeezing of the output field.Physical Review A 01/2011; 05381750. · 3.04 Impact Factor - SourceAvailable from: Hichem Eleuch[Show abstract] [Hide abstract]
ABSTRACT: We study nonlinear effects in an optomechanical system containing a quantum well. The nonlinearity due to the optomechanical coupling leads to a bistable behavior in the mean intracavity photon number and substantial squeezing in the transmitted field. We show that the optical bistability and the degree of squeezing can be controlled by tuning the power and frequency of the pump laser. The transmitted field intensity spectrum consists of six distinct peaks corresponding to optomechanical, polariton, and hybrid resonances. Interestingly, even though the quantum well and the mechanical modes are not directly coupled, their interaction with the common quantized cavity mode results in appearance of hybrid resonances.Physical Review A 04/2012; 8550. · 3.04 Impact Factor - SourceAvailable from: Hichem Eleuch[Show abstract] [Hide abstract]
ABSTRACT: We investigate nonlinear effects in an electromechanical system consisting of a superconducting charge qubit coupled to a transmission line resonator and a nanomechanical oscillator, which in turn is coupled to another transmission line resonator. The nonlinearities induced by the superconducting qubit and the optomechanical coupling play an important role in creating optomechanical entanglement as well as the squeezing of the transmitted microwave field. We show that strong squeezing of the microwave field and robust optomechanical entanglement can be achieved in the presence of moderate thermal decoherence of the mechanical mode.We also discuss the effect of the coupling of the superconducting qubit to the nanomechanical oscillator on the bistability behavior of the mean photon number.Physical Review A 01/2014; 89:013841. · 3.04 Impact Factor
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PHYSICAL REVIEW A 83, 023822 (2011)
External-field effect on quantum features of radiation emitted by a quantum well in a microcavity
Eyob A. Sete*and Sumanta Das
Institute for Quantum Science and Engineering and Department of Physics and Astronomy, Texas A&M University,
College Station, Texas 77843-4242, USA
H. Eleuch
Department of Physics and Astronomy, College of Science, P. O. Box 2455, King Saud University, Riyadh 11451, Saudi Arabia
(Received 15 December 2010; published 25 February 2011)
Weconsiderasemiconductorquantumwellinamicrocavitydrivenbycoherentlightandcoupledtoasqueezed
vacuum reservoir. By systematically solving the pertinent quantum Langevin equations in the strong-coupling
and low-excitation regimes, we study the effect of exciton-photon detuning, external coherent light, and the
squeezed vacuum reservoir on vacuum Rabi splitting and on quantum statistical properties of the light emitted by
the quantum well. We show that the exciton-photon detuning leads to a shift in polariton resonance frequencies
and a decrease in fluorescence intensity. We also show that the fluorescent light exhibits quadrature squeezing,
which predominately depends on the exciton-photon detuning and the degree of the squeezing of the input field.
DOI: 10.1103/PhysRevA.83.023822 PACS number(s): 42.55.Sa, 78.67.De, 42.50.Dv, 42.50.Lc
I. INTRODUCTION
Study of radiation-matter interaction in two-level quantum
mechanicalsystemshaveledtoseveralfascinatingphenomena
like the Autler-Townes doublet [1], vacuum Rabi splitting
[2–4], antibuching, and squeezing [5–7]. In particular, in-
teraction of two-level atoms in a cavity with a coherent
source of light and coupled to a squeezed vacuum has been
extensivelystudied[8–16].Currently,thereisrenewedinterest
insuchstudiesfromthecontextofsemiconductorsystemslike
quantum dots (QDs) and wells (QWs) [17–21], given their
potential application in opto-electronic devices [22]. In this
regard, intersubband excitonic transitions which have similar-
ities to two-level atomic systems has been primarily exploited.
However, itisimportanttounderstand thatthequantum nature
of fluorescent light emitted by excitons in QWs embedded
inside a microcavity somewhat differs from that of atomic
cavity QED predictions. For example, unlike antibunching
observed in atoms embedded in a cavity, a QW exhibits
bunching effects in the fluorescent spectrum of the emitted
radiation [23,24]. Further, in the strong-coupling regime,
for a resonant microcavity-QW interaction, exciton-photon
mode splitting and oscillatory excitonic emission has been
demonstrated [25–27]. Recently, the effect of a nonresonant
strong drive on the intersubband excitonic transition has been
investigated and observation of Autler-Townes doublets was
reported [28–30]. In light of these new results, an eminent
question of interest is thus: How is the quantum nature of
radiation emitted from a QW in a microcavity affected in the
presence of a squeezed vacuum environment and nonresonant
drive? We investigate this in the current paper.
We explore the interaction between an external field and a
QW placed in a microcavity coupled to a squeezed vacuum
reservoir. Our analysis is restricted to the weak-excitation
regime where the density of exciton is small. This allows us to
neglect any exciton-exciton interaction, thereby simplifying
our problem considerably and yet preserving the physical
*yob a@yahoo.com
insight. We further assume the cavity-exciton interaction to
be strong, which brings in interesting features. Note that
we consider the external field to be in resonance with the
cavity mode throughout the paper. We analyze the effect
of exciton-photon detuning, external coherent light, and the
squeezed reservoir on the quantum statistical properties and
polaritonresonancesinthestrong-couplingandlow-excitation
regimes. The effect of the coherent light on the behavior
of the dynamical evolution of the intensity fluorescent light
is remarkably different from that of the squeezed vacuum
reservoir due to the nature of photons generated by the two
systems. This is due to the distinct nature of the photons
generated by the coherent and squeezed inputs. This effect
is manifested on the intensity of the fluorescent light. Both
sources lead to excitation of two or more excitons in the
quantum well creating a probability for emission of two or
more photons simultaneously. As a result of this, the photons
tend to propagate in bunches other than at random. Moreover,
the fluorescent light emitted by exciton in the quantum well
exhibitsanonclassicalproperty,namelyquadraturesqueezing.
II. MODEL AND QUANTUM LANGEVIN EQUATIONS
We consider a semiconductor QW in a cavity driven by
external coherent light and coupled to a single-mode squeezed
vacuum reservoir. The scheme is outlined in Fig. 1. In this
work, we are restricted to a linear regime in which the density
of excitons is small so that exciton-exciton scattering can be
ignored. We assume that the driving laser is at resonance with
the cavity mode while the exciton transition frequency is of
resonantwiththecavitymodebyanamount? = ω0− ωcwith
ω0andωcbeingtheexciton-andcavity-modefrequencies.The
interaction of the cavity mode with the resonant pump field
and the exciton is described, in the rotating-wave and dipole
approximations, by the Hamiltonian
H = ?b†b + iε(a†− a) + ig(a†b − ab†) + Hloss,
where a and b are the annihilation operators for the cavity and
exciton modes satisfying the commutation relation [a,a†] =
(1)
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EYOB A. SETE, SUMANTA DAS, AND H. ELEUCHPHYSICAL REVIEW A 83, 023822 (2011)
FIG. 1. (Color online) Schematic representation of a QW in a
driven cavity coupled to a squeezed vacuum reservoir.
[b,b†] = 1 respectively; g is the photon-exciton coupling
constant; ε, assumed to be real and constant, is proportional to
the amplitude of the pump field, and Hlossis the Hamiltonian
that describes the interaction of exciton with the vacuum
reservoir and also the interaction of the cavity mode with the
squeezed vacuum reservoir.
The quantum Langevin equations, taking into account the
dissipation processes, can be written as
da
dt
= −κ
?γ
2a + gb + ε + F(t),
?
(2)
db
dt
= −
2+ i?b − ga + G(t),
(3)
where κ and γ are cavity-mode decay rate via the port
mirror and spontaneous-emission decay rate for the exciton,
respectively, and F =√κFinand G =√γGinwith Finand
Gin being the Langevin noise operators for the cavity and
exciton modes, respectively. Both noise operators have zero
mean, that is, ?Fin? = ?Gin? = 0. For a cavity mode coupled to
asqueezedvacuumreservoir,thenoiseoperatorFin(t)satisfies
the following correlations:
?Fin(t)F†
?F†
in(t?)? = (N + 1)δ(t − t?),
in(t)Fin(t?)? = Nδ(t − t?),
?Fin(t)Fin(t?)? = ?F†
where N = sinh2r and M = sinhr coshr with r being the
squeeze parameter characterize the mean photon number and
the phase correlations of the squeezed vacuum reservoir,
respectively. Further, the exciton noise operator Ginsatisfies
the following correlations:
(4)
(5)
in(t)F†
in(t?)? = Mδ(t − t?),
(6)
?Gin(t)G†
in(t?)? = δ(t − t?),
(7)
?G†
in(t)Gin(t?)? = ?Gin(t)Gin(t?)? = ?G†
Following the method outlined in [21], we obtain the
solution of the quantum Langevin equations (2) and (3) in
the strong coupling regime (g ? γ,κ) to be
a(t) = η1(t)ε + η+(t)a(0) + η3(t)b(0)
+
0
in(t)G†
in(t?)? = 0. (8)
?t
dt?η+(t − t?)F(t?) +
?t
0
dt?η3(t − t?)G(t?), (9)
b(t) = η−(t)b(0) − η4(t)ε − η3(t)a(0)
−
0
where
η1(t) =1
1
2µ[2µcosµt ± i?sinµt]e−(?+i?/2)t,
η3(t) =g
η4(t) =1
g
?t
dt?η3(t − t?)F(t?) +
?t
0
dt?η−(t − t?)G(t?),
(10)
µsinµte−(?+i?/2)t,
(11)
η±(t) =
(12)
µsinµte−(?+i?/2)t,
cosµt +i?
(13)
g−1
?
2µsinµt
?g2+ ?2/4. Using these
?
e−(?+i?/2)t,
(14)
where ? = (κ + γ)/4 and µ =
solutions, we study the dynamical behavior of intensity, power
spectrum, second-order correlation function, and quadrature
variance for the fluorescent light emitted by the quantum well
in the following sections.
III. INTENSITY OF FLUORESCENT LIGHT
In this section, we analyze the properties of the fluorescent
lightemittedbyexcitonsinthequantumwell.Inparticular,we
study the effect of the external driving field, exciton-photon
detuning, and the squeezed vacuum reservoir. Note that the
intensity of the fluorescent light is proportional to the mean
numberofexcitons.Tothisend,theintensityofthefluorescent
light can be expressed in terms of the solution of the Langevin
equations as
?b†b? = ε2|η4(t)|2+ |η−(t)|2+ κN
Here we have assumed the cavity mode is initially in vac-
uum state [?a†(0)a(0)? = 0] while the quantum well initially
contains only one exciton [?b†(0)b(0)? = 1]. In (15), the
first term corresponds to the contribution from the external
coherent light while the last term is due to the the squeezed
vacuum reservoir. It is also easy to see that the intensity is
independentoftheparameterM,whichcharacterizesthephase
correlations of the reservoir, implying that the same result
could be obtained if the cavity mode is coupled to a thermal
reservoir.
Performing the integration using Eqs. (13) and (14), we
obtain
?b†b? =ε2
+ε2
where
?2
4µ2sin2µt + cos2µt,
λ2(t) = −g2
4µ2
λ3(t) = cosµt cos(?t/2) +?
?t
0
|η3(t − t?)|2dt?.
(15)
g2+g2κN
4?µ2+ (λ1+ κNλ2)e−2?t
g2(λ1e−2?t− 2λ3e−?t),
(16)
λ1(t) =
(17)
?1
?+1
µsin2µt
?
,
(18)
2µsinµt sin(?t/2). (19)
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PHYSICAL REVIEW A 83, 023822 (2011)
5
9
7
r
0
2
02468
2
4
6
8
γ t
b†b
FIG. 2. (Color online) Plots of the fluorescent intensity [Eq. (21)]
vs scaled time γt for κ/γ = 1.2, g/γ = 5, and ?/γ = 2 and for
different values of ε/γ.
We immediately see that the intensity of the fluorescent light
reduces in the steady state to
?b†b?ss=ε2
g2+g2κN
4?µ2.
(20)
Asexpected,thesteady-stateintensityisinverselyproportional
to the decay rates. The higher the decay rate, the lower the
intensity becomes, and vice versa.
In order to clearly see the effect of the external coherent
light on the intensity, we set N = 0 in Eq. (20) and obtain
?b†b? =ε2
Figure 2 shows the dependence of the intensity of the
fluorescent light on the external coherent field. In general, the
intensity increases with the amplitude of the pump field and
exhibits nonperiodic damped oscillations. Although there is a
decreaseinthemeannumberofexcitonsfortheinitialmoment,
cavity photons gradually excite one or more excitons in the
quantum well, leading to enhanced emission of fluorescence.
However, the excitation of excitons saturates as time progress
is limited by the strength of the applied field. From the
steady-state intensity, ε2/g2, we easily see that the field
strength has to exceed the exciton-photon coupling constant in
order to see more than one exciton in the quantum well in the
long time limit.
Ontheotherhand,theeffectofthesqueezedvacuumcanbe
studied by turning off the external driving field. Thus setting
ε = 0 in Eq. (20), we get
?b†b? =g2κN
InFig.3,weplottheintensityofthelightemittedbytheexciton
[Eq. (22)] as a function of scaled time γt for a given photon-
exciton detuning. This figure illustrates the dependence of the
intensity on the photons generated from the squeezed vacuum
reservoir and impinging on the cavity through the partially
transmitting mirror. Here also, the intensity exhibits damped
oscillations at frequency 2µ = 2(g2+ ?2/4)1/2, indicating
exchange of energy between the excitons and cavity mode.
Even though the intensity decreases at the initial moment,
it gradually increases, showing oscillatory behavior, to its
g2(1 + λ1e−2?t− 2λ3e−?t).
(21)
4?µ2+ (λ1+ κNλ2)e−2?t.
(22)
r
1.2
r
1.5
r
1.8
0
2
02468
1
2
3
4
γ t
b†b
FIG. 3. (Coloronline)Plotsofthefluorescentintensity[Eq.(22)],
vs scaled time γt for κ/γ = 1.2, g/γ = 5, ?/γ = 2, no external
driving field (ε/γ = 0), and for different values of r.
steady-state value. Unsurprisingly, the intensity increases
with the number of photons coming in through the mirror.
Comparing Figs. 2 and 3, we note that the intensity has
different behaviors for the two cases. This can be explained
in terms of the nature of photon each source is producing. In
the case of coherent light, the photon distribution is Poisson
and the photons propagates randomly. This leads to uneven
excitationofexcitonsthatresultsinthenonperiodicoscillatory
natureoftheintensity.Forthecaseofsqueezedvacuumsource,
however, the photons show the bunching property and hence
can excite two or more excitons at the same time. This in turn
implies a steady increase in the mean number of excitons in
the quantum well, depending on the strength of the impinging
squeezed vacuum field.
For the sake of completeness, we further consider the effect
of detuning on the intensity of the fluorescent light at a given
pump field strength and squeezed field. Figure 4 shows the
intensity as a function of scaled time γt. When the photon
is out of resonance with the exciton frequency, there will be
fewer excitons in the quantum well, and hence the fluorescent
intensity decreases. This is clearly shown in Fig. 4.
4
0
0246810
2
4
6
8
γt
b†b
FIG. 4. (Color online) Plots of the fluorescent intensity [Eq. (16)]
vs scaled time γt for κ/γ = 1.2, g/γ = 5, r = 1.8, and pump
amplitude ε/γ = 7 and for different values of exciton-photon
detuning (?/γ).
023822-3
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EYOB A. SETE, SUMANTA DAS, AND H. ELEUCHPHYSICAL REVIEW A 83, 023822 (2011)
3
0
1050510
0.00
0.05
0.10
0.15
0.20
0 γ
Sincohω
FIG. 5. (Color online) Plots of the incoherent component of the
powerspectrum[Eq.(26)]vsscaledfrequency(ω − ω0)/γ forκ/γ =
1.2, g/γ = 6, and squeeze parameter r = 1 and for different values
of detuning, ?/γ.
IV. POWER SPECTRUM
The power spectrum of the fluorescent light in the steady
state is given by
S(ω) =1
πRe
?∞
0
dτei(ω−ω0)τ?b†(t)b(t + τ)?ss,
(23)
where ss stands for steady state. The two-time correlation
function that appears in the above integrand is found to be
?b†(t)b(t + τ)?ss=ε2
g2+κNg2
×(µcosµτ + ? sinµτ).
4?µ3e−(?+2i?/2)τ
(24)
Now employing this result in (23), performing the resulting
integration, and carrying out the straightforward arithmetic,
we obtain
S(ω) =
ε2
2πg2δ(ω − ω0) + Sincoh(ω),
(25)
where
Sincoh(ω) =
kNg2
16πµ3
?
? + 4µ − 2ω
?2+??
−? + 4µ + 2ω
?2+??
2+ µ − (ω − ω0)?2
2− µ − (ω − ω0)?2
+
?
,
(26)
where ? = (κ + γ)/4. We note that the power spectrum has
two components: coherent and incoherent parts. The coherent
component is represented by the δ function, which indeed
corresponds to the coherent light. The incoherent component
given by (26) arises as a result of the squeezed photons
coming through the port mirror. From Eq. (26), it is clear
that the spectrum of the incoherent light is composed of
two Lorentzians having the same width ? but centered at
two different frequencies: ω − ω0= µ + ?/2 and ω − ω0=
µ − ?/2. We then see that the detuning leads to a shift in the
resonance frequency components observed in zero detuning
(? = 0).
In Fig. 5, we plot the incoherent component of the power
spectrum as a function of scaled time γt for the cavity mode
initially in vacuum state and for the quantum well initially
containingoneexciton.Forzerodetuning,thepowerspectrum
consists of two well-resolved peaks centered at ω − ω0= ±g.
Thissplittingcanbeunderstoodfromthedressed-stateenergy-
level diagram [see Fig. 6(a)]. Note that for the case in which
there is only one excitation, there are two possible degenerate
bare states: |1,0?, one exciton and no photon; and |0,1?, one
photon no exciton. However, the strong exciton-photon cou-
pling lifts the degeneracy of these two bare states and results
in two dressed states (polaritons): |+? = (|1,0? + i|0,1?)/√2
and |−? = (|1,0? − i|0,1?)/√2 with eigenvalues g and −g,
respectively. In general, since the exciton-photon system is
coupled to the environment, polaritons are unstable states.
Thus, the decay of the exciton and cavity photon leads to
polariton’s decay, which yields two peaks in the emission
spectrum.
FIG. 6. (Color online) (a) Dressed-state energy-level diagram for single- and two-excitation manifolds when the exciton is at
resonance with photon. (b) Dressed-states energy-level diagram when the exciton frequency is detuned by ? from that of the photon. We have
assumed ? to be positive for the sake of simplicity. The bare states |n,m?(n,m = 0,1,2) represent n numbers of excitons and m numbers of
photons. Even though there are six possible transitions, there are only two distinct transition frequencies, namely: ω − ω0= µ + ?/2 and
ω − ω0= −µ + ?/2, where µ =
?g2+ ?2/4.
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EXTERNAL-FIELD EFFECT ON QUANTUM FEATURES OF ...
PHYSICAL REVIEW A 83, 023822 (2011)
TABLE I. List of eigenvalues and eigenstates for single- and two-excitation manifolds. Here χ±=
?4g2+ (? ± 2µ)2.
Eigenstates (polaritons)Eigenvalues (shifts)
Single-excitation
manifold
?/2 + µ
|1?+= [(? + 2µ)|1,0? + 2ig|0,1?]/χ+
?/2 − µ
? + 2µ
|1?−= [(? + 2µ)|1,0? + 2ig|0,1?]/χ−
|2?+= [−i√2g|2,0? + (? + 2µ)|1,1? + i√2g|0,2?]/χ+
|2?0= [−i√2g|2,0? + ?|1,1? + i√2g|0,2?]/µ
|2?−= [−i√2g|2,0? + (? − 2µ)|1,1? + i√2g|0,2?]/χ−
Two-excitation
manifold
?
? − 2µ
It is worth noting that even though we start off with a
single exciton in the quantum well, the cavity photons excite
two or more excitons in the quantum well. This results in
moredressedstatesinmultiexcitationmanifolds.Forexample,
as shown in Fig. 6(a), for two-excitation manifolds there are
three dressed states which are equally spaced in energy. This
energy separation is the same as the energy separation in the
one-excitation manifold. Out of the six possible transitions,
from two excitations to single excitation and then from single
to ground states, there are only two distinct frequencies.
Therefore,theemissionspectrumconsistsoftwopeaks.Thisis
different from the atom-photon coupling in which the increase
in excitation number will increase the number of emission
spectrum peaks.
On the other hand, for the nonzero detuning case, the
emission spectrum has two peaks whose centers are shifted
to red (for positive detuning). Here, the one-excitation bare
states (|1,0?,|0,1?) are separated by ? and the two-excitation
states (|2,0?,|1,1?, and |0,2?) as well. The eigenvalues and
corresponding eigenstates are given in Table I. The exciton-
photon coupling leads to the generation of dressed states
(polaritons). The decay of these states to the one-excitation
22
g22
4
1050510
6
4
2
0
2
4
6
0 γ
γ
FIG. 7. (Color online) Density plot of the incoherent component
ofthepowerspectrum[Eq.(26)]vsscaledfrequency(ω − ω0)/γ and
?/γ for κ/γ = 1.2 and g/γ = 6 and for squeeze parameter r = 1.
state and to the ground state give rise to two emission
peaks whose frequencies are different from the zero detuning
case as shown in Fig. 6(b). Further, the density plot for
the power spectrum clearly shows that there are indeed
two peaks separated by 2µ = 2?g2+ ?2/4 as illustrated in
Fig. 7.
V. SECOND-ORDER CORRELATION FUNCTION
In this section, we study the second-order correlation func-
tion of the light emitted by the quantum well. Second-order
correlation function is a measure of the photon correlations
between some time t and a later time t + τ. It is also an
indicator of a quantum feature that does not have a classical
analog. Quantum mechanically, the second-order correlation
function is defined by
g(2)(τ) =?b†(t)b†(t + τ)b(t + τ)b(t)?ss
?b†(t)b(t)?2
The correlation function that appears in (27) can be obtained
using the solution (20) together with the properties of the
Langevin noise forces. Applying the Gaussian properties of
the noise forces together with Eqs. (4)–(6) and (20), we obtain
?κ2
+κε2
ss
.
(27)
g(2)(τ) = 1 +
1
?b†b?2
g2[MA1+ NA2cos(?τ/2)]e−?τ
ss
4
?4M2A3+ N2A2
2
?e−2?τ
?
,
(28)
where
A1(τ) =2µcosµτ sin(?τ/2) − ?cos(?τ/2)sinµτ
+g2cosµτ[2? cos(?τ/2) − ?sin(?τ/2)]
µ2(?2+ 4?2)
g2
2?µ3(µcosµτ + ? sinµτ),
4µ2
, (29)
A2(τ) =
(30)
A3(τ) =4µ2cos2µτ + ?2sin2µτ + 4?µsin(2µτ)
16µ2(?2+ 4?2)
and?b†b?ssisgivenby(20).ThefirstterminEq.(28)represents
the second-order correlation function for the coherent light.
,
(31)
023822-5
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EYOB A. SETE, SUMANTA DAS, AND H. ELEUCHPHYSICAL REVIEW A 83, 023822 (2011)
0
r
1
0
4
0.00.51.01.5
γ τ
2.02.53.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
g2τ
FIG. 8. (Color online) Plot of second-order correlation function
vs normalized time γτ for g/γ = 5, κ/γ = 1.2, ?/γ = 0, and
squeezingparameterr = 1andfordifferentvalueofpumpamplitude
ε/γ.
ThiscaneasilybeseenbysettingN = M = 0.InEq.(28),the
firstterminsidethebigsquarebracketisthecontributiontothe
second-order correlation function from the squeezed vacuum
reservoir while the second term describes the interference
between the coherent field and the reservoir. Note that at
τ → ∞ g(2)becomes unity, as it should be, showing no
correlation between the photons.
The dynamical behavior of the second-order correlation
function is illustrated in Fig. 8. We see from this figure
that the correlation function shows oscillatory behavior with
oscillation frequency equal to the photon-exciton coupling
constant(g)forthezero-detuningcaseandintheabsenceofthe
external driving field. However, the frequency of oscillation is
reduced by a factor of 1/2 in the presence of an external
coherent field.
It is easy to see that g(2)(0) is always greater than unity,
indicatingphotonbunching.Thisisincontrasttowhathasbeen
observed in atomic cavity QED, where the photons show an
antibunchingproperty[16].Thisisduetothefactthatthereisa
finite time delay between absorbtion and subsequent emission
of a photon by the atom. In the case of semiconductor cavity
QED, however, the cavity photons can excite two or more
excitonsatthesametime,dependingonthenumberofphotons
in the cavity, leading to possible multiphoton emission. This is
the reason why the photons emitted by excitons are bunched.
Indeed,excitationoftwoormoreexcitonsinthequantumwell
is shown in Figs. 2–4.
VI. QUADRATURE SQUEEZING
Next, we study the squeezing properties of the fluorescent
light by evaluating the variances of the quadrature operators.
The variances of the quadrature operators for the fluorescent
light are given by
?b2
±= 1 + 2?b†b? ± [?b2? + ?b†2?] ∓ (?b†? ± ?b?)2,
where b+= (b†+ b) and b−= i(b†− b). It is easy to show
using these definitions that the quadrature operators satisfythe
commutation relation [b+,b−] = 2i. It is well known that for
the fluorescent light to be in a squeezed state, the variances
of the quadrature operators should satisfy the condition that
(32)
either?b2
of the noise operators, we find the variances to be
+< 1or?b2
−< 1.UsingEq.(10)andtheproperties
?b2
±= 1 +κNg2
+[2λ1(t) + 2κNλ2(t) ± κMλ4]e−2?t,
inwhichλ1andλ2aregivenbyEqs.(17)and(18),respectively
and
λ4=g2
µ2
µ2(?2+ 4?2)
−sin[(? − 2µ)t]
2(? − 2µ)
It is straightforward to see that in the steady state the variances
reduce to
2?µ2±
2κMg2?
µ2(?2+ 4?2)
(33)
??sin(?t/2) − 2? cos(?t)
−sin[(? + 2µ)t]
2(? + 2µ)
?
.
?b2
+= 1 +κNg2
2?µ2+
2κMg2?
µ2(?2+ 4?2),
2κMg2?
µ2(?2+ 4?2).
(34)
?b2
−= 1 +κNg2
2?µ2−
(35)
Fromtheaboveexpressionswefindthat,inthesteadystate,
the quadrature variances crucially depend on the detuning, the
cavity-exciton coupling strength, and amount of squeezing
provided by the reservoir. Further, it is also apparent that
if there is any squeezing it can only be present in the b−
quadrature. Thus, for rest of this section, we discuss only the
properties of variance in the b−quadrature. As a special case,
we consider that the cavity mode to be at resonance with with
the excitonic transition frequency and put ? = 0 in Eq. (35).
We then find that
κ
κ + γ(1 − e−2r) < 1.
Equation (36) then suggests that higher squeezing in the
reservoir leads to better squeezing of the fluorescent light.
In Fig. 9, we confirm this behavior by plotting the steady
state ?b2
from Eq. (36) we see that as e−2r→ 0 quickly with increase
in r, the maximum possible squeezing achievable in our
system is 50% for κ = γ. This is also depicted to be true in
Fig. 9.
?b2
−= 1 −
(36)
−as a function of the squeezing parameter r. Further,
0
0.5
0.00.20.40.60.8
r
1.01.21.4
0.6
0.8
1.0
1.2
1.4
b2
FIG. 9. (Color online) Plots of the steady-state quadrature vari-
ance [(35)] vs squeeze parameter r for g/γ = 5 and κ/γ = 1.2 and
for the different values of exciton-photon detuning ?/γ.
023822-6
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EXTERNAL-FIELD EFFECT ON QUANTUM FEATURES OF ...
PHYSICAL REVIEW A 83, 023822 (2011)
1.0
0.5
0.0
r
1.0
0246810
0.5
1.0
1.5
2.0
2.5
3.0
γ t
b2
FIG. 10. (Color online) Plots of the quadrature variance [(33)]
vs scaled time γt for g/γ = 5, κ/γ = 1.2, and squeeze parameter
r = 1 and for the different values of exciton-photon detuning ?/γ.
In the presence of detuning (? ?= 0), the behavior of ?b2
changes dramatically. We find that for some small detuning
?/γ = 0.5 there exists a range of the squeezing parameter
r(0 < r < 1.3)whereonecanseesqueezingofthefluorescent
light emitted from the exciton; however, for higher value of
r it vanishes. This thus implies that in presence of detuning
stronger squeezing of the reservoir leads to negative effects
on the squeezing of the emitted radiation from the exciton.
In Fig. 10, we plot the time evolution of the quadrature
variance ?b2
time γt for r = 1 and different values of detuning. It is
seen, in general, that the variance oscillates initially with
the amplitude of oscillation gradually damping out at longer
time. Eventually, at large enough time, the variance becomes
flat and approaches the steady-state value. Interestingly, our
results show that even though there is no squeezing of the
fluorescent light at the initial moment, for small or zero
detuning, transient squeezing gradually develops. Moreover,
we also find that for weak squeezing of the reservoir, even in
the presence of small detuning, the initial transient squeezing
is sustained and finally leads to a steady-state squeezing. This
can be understood as a consequence of strong interaction of
−
−[Eq. (33)] as a function of the normalized
the quantum well with the squeezed photon entering via the
cavity mirror. In case of large detuning, the exciton is unable
to absorb photons from the squeezed reservoir, and thus no
squeezing develops in the fluorescence.
VII. CONCLUSION
In this paper, we consider a semiconductor quantum well
in a cavity driven by external coherent light and coupled to a
single-mode squeezed vacuum reservoir. We study the photon
statistics and nonclassical properties of the light emitted by
the quantum well in the presence of exciton-photon detuning
in the strong-coupling regime. The effects of coherent light
and the squeezed vacuum reservoir on the intensity of the
fluorescencearequitedifferent.Theformerleadstoatransient
peak intensity, which eventually decreases to a considerably
smaller steady-state value. In contrast, the latter, however,
gives rise to a gradual increase in the intensity and leads
to maximum intensity at steady state. This difference is
attributed to the nature of photons that the two sources
produce. As a signature of strong coupling between the
excitons in the quantum well and cavity photons, the emission
spectrum consists of two peaks corresponding to the two
eigenenergies of the dressed states. Further, we find that
the fluorescence exhibits a nonclassical feature—quadrature
squeezing—as a result of strong interaction of the excitons
with the squeezed photons entering via the cavity mirror. In
view of recent successful experiments on the Autler-Townes
effect in GaAs/AlGaAs [30] and gain without inversion
in semiconductor nanostructures [31], the quantum statistical
propertiesofthefluorescenceemittedbythequantumwellcan
be tested experimentally.
ACKNOWLEDGMENT
One of the authors (E.A.S.) is supported by the Herman
F. Heep and Minnie Belle Heep Texas A&M University
Endowed Fund held and administered by the Texas A&M
Foundation.
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