# 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.

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**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: Eyob A. Sete[Show abstract] [Hide abstract]

**ABSTRACT:**We investigate nonlinear effects in an electromechanical system involving a superconducting charge qubit coupled to 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 generating 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. We also discuss the effect of the coupling of the superconducting qubit to the nanomechanical oscillator on the bistability behaviour of the mean photon number.12/2013;

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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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EXTERNAL-FIELD EFFECT ON QUANTUM FEATURES OF ...

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).

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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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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)

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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.

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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.

[1] S. H. Autler and C. H. Townes, Phys. Rev. 100, 703 (1955).

[2] G. S. Agarwal, Phys. Rev. Lett. 53, 1732 (1984).

[3] R. J. Thompson, G. Rempe, and H. J. Kimble, Phys. Rev. Lett.

68, 1132 (1992).

[4] G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer,

Nature Physics 2, 81 (2006).

[5] C. W. Gardiner, Phys. Rev. Lett. 56, 1917 (1986).

[6] H. J. Carmichael, A. S. Lane, and D. F. Walls, Phys. Rev. Lett.

58, 2539 (1987); J. Mod. Opt. 34, 821 (1987).

[7] R. Vyas and S. Singh, Phys. Rev. A 45, 8095 (1992).

[8] N. Ph. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and

A. S. Parkins, Phys. Rev. Lett. 75, 3426 (1995).

[9] G. S. Agarwal, Phys. Rev. A 40, 4138 (1989).

[10] A. S. Parkins, Phys. Rev. A 42, 4352 (1990).

[11] J. I. Cirac and L. L. Sanchez-Soto, Phys. Rev. A 44, 1948

(1991).

[12] P. R. Rice and C. A. Baird, Phys. Rev. A 53, 3633 (1996).

[13] W. S. Smyth and S. Swain, Phys. Rev. A 53, 2846 (1996).

[14] J. P. Clemens, P. R. Rice, P. K. Rungta, and R. J. Brecha, Phys.

Rev. A 62, 033802 (2000).

[15] C. E. Strimbu, J. Leach, and P. R. Rice, Phys. Rev. A 71, 013807

(2005).

[16] E. Alebachew and K. Fessah, Opt. Commun. 271, 154 (2007);

E. Alebachew, J. Mod. Opt. 55, 1159 (2008).

[17] A. Baas, J. Ph. Karr, H. Eleuch, and E. Giacobino, Phys. Rev. A

69, 023809 (2004).

[18] A. Quattropani and P. Schwendimann, Phys. Status Solidi 242,

2302 (2005).

[19] H. Eleuch, J. Phys. B 41, 055502 (2008).

[20] H. Eleuch, Eur. Phys. J. D 49, 391 (2008); 48, 139 (2008).

[21] E. A. Sete and H. Eleuch, Phys. Rev. A 82, 043810

(2010).

[22] A. J. Shields, Nat. Photonics 1, 215 (2007).

[23] R. Vyas and S. Singh, J. Opt. Soc. Am. B 17, 634 (2000).

[24] D. Erenso, R. Vyas, and S. Singh, Phys. Rev. A 67, 013818

(2003).

023822-7

Page 8

EYOB A. SETE, SUMANTA DAS, AND H. ELEUCHPHYSICAL REVIEW A 83, 023822 (2011)

[25] C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, Phys.

Rev. Lett. 69, 3314 (1992).

[26] S. Pau, G. Bjork, J. Jacobson, H. Cao, and Y. Yamamoto, Phys.

Rev. B 51, 14437 (1995); S. Pau et al., Appl. Phys. Lett. 66,

1107 (1995).

[27] J. Jacobson, S. Pau, H. Cao, G. Bjork, and Y. Yamamoto, Phys.

Rev. A 51, 2542 (1995).

[28] J. F. Dynes, M. D. Frogley, M. Beck, J. Faist, and C. C. Phillips,

Phys. Rev. Lett. 94, 157403 (2005).

[29] S. G. Carter et al., Science 310, 651 (2005).

[30] M. Wagner, H. Schneider, D. Stehr, S. Winnerl, A. M. Andrews,

S. Schartner, G. Strasser, and M. Helm, Phys. Rev. Lett. 105,

167401 (2010).

[31] M. D. Frogley et al., Nat. Mater. 5, 175 (2006).

023822-8