Analytical results from the quantum theory of a single-emitter nanolaser. II

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by  Nikolay V. Larionov

Analytical results from the quantum theory of a single-emitter nanolaser. II.
Nikolay V. Larionov1∗ and Mikhail I. Kolobov2
1Department of Theoretical Physics, St-Petersburg State Polytechnic University, 195251, St.-Petersburg, Russia and
2Laboratoire de Physique des Lasers, Atomes et Molécules,
Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France.
(Dated: January 25, 2013)
Using two equivalent approaches, Heisenberg-Langevin and density operator, we investigate the
properties of nanolaser: an incoherently pumped single two-level system interacting with a single-
cavity mode of finite finesse. We show that in the case of good-cavity regime the Heisenberg-
Langevin approach provides the analytical results for linewidth, amplitude fluctuation spectrum,
3
andintracavityMandelQ-parameter. Inthebad-cavityregime weestimatethefrequencyatwhich
1
thepeakofrelaxationoscillationsappears. Withthehelpofthemasterequationfordensityoperator
0
written in terms of coherent states we obtain approximated expression for Glauber-Sudarshan P
2
function. This solution can be used in the case of good-cavity regime and allows us to investigate

n in more detail the thresholdless behaviour of the nanolaser. All two approaches are in a very good
a agreement with each other and numerical simulations.
J

3
2 1. INTRODUCTION latter we will prove our previous results, analyze some
limited cases and threshold behaviour of nanolaser.
]
s
One of the first theoretical work devoted to possibil-
c
i ity of realization of a single-emitter laser was published 2. HEISENBERG-LANGEVIN APPROACH
t
p by Mu and Savage in [1]. Three- and four-level pumped
o emittersplacedintolossycavityandinteractedwithcav-
2.1 Model. Semiclassical theory
.
s ity mode have been considered there. Some interesting
c effects which do not appear in conventional lasers have
i Thesimplestmodelofasingle-emitternanolaseristhe
s been revealed, such as squeezing, self-quenching in in-
y singletwo-levelsystemplacedinsidethesingle-modecav-
coherently pumped laser,deviations from the Schawlow-
h ityandincoherentlypumpedtoitsupperlevel. Justfour
Townes formula for the linewidth. Subsequent publica-
p constants characterize this nanolaser: Γ – the incoher-
[ tions relatedto single-emitterproblem[2–9]haveproven
ent pumping rate, pumping process associated with the
these results and also discovered new effects connected
1 transition 1 2 ; γ/2 – the decay rate of polariza-
with the presence of one emitter: vacuum-Rabi doublet | i → | i
v tion due to spontaneous emission to modes other than
2 in the spectrum [2, 4], lasing without inversion [6], en-
the laser mode; κ/2 – the field decay rate in the cavity;
5 tanglement between emitter and field subsystems [7]. In
g – the coupling constantbetween the field and the two-
6 mostoftheseworksresultshavebeenobtainedwithhelp
5 levelsystem. TheHeisenberg-Langevinequationsforour
ofdifferenttypeofnumericalmethodsappliedtothemas-
. nanolaser are
1 ter equation for density operator or to the Heisenberg-
0 Langevin equations. daˆ(t) = κaˆ(t)+gσˆ(t)+fˆ(t),
3 dt −2 a
1 Nowadays a single-emitter laser is realized in exper-
d 1
: iments where the role of emitter can play the single σˆ(t) = (Γ+γ)σˆ(t)+gDˆ(t)aˆ(t)+fˆ (t),
v dt −2 σ
atom [10], or ion [11] or quantum dot [12]. In this con-
Xi nection,thepurposeofourpaperistoprovideanalytical dDˆ(t) = Γ γ (Γ+γ)Dˆ(t)
dt − −
r results which can be considered as a tools for experi-
a menter. ThusinourpreviousBriefReport[13]withhelp 2g σˆ†(t)aˆ(t)+aˆ†(t)σˆ(t) +fˆ (t). (1)
D
−
of the Heisenberg-Langevin approach we obtained ana-
h i
lytical expressions for linewidth, amplitude fluctuation Here aˆ(t) and aˆ†(t) are the photon annihilation and cre-
spectrum and Mandel Q parameter, which describe the ation operators in the cavity mode, σˆ = 1 2 is the
| ih |
behaviour of a single-emitter laser in the case of good operator of polarization of the two-level system, and
cavity regime. We also improved the condition required Dˆ = 2 2 1 1 is the operator of inversion between
| ih |−| ih |
for thresholdless regime. In this paper in Sec. 2 we will the upper level 2 and the lower level 1 of the sys-
| i | i
considerandanalyzeinmoredetailthelinearizationpro- tem. We use the fact that for a single two-level system
cedurewhichweusedearlierandextendourcalculations 2 2 + 1 1 =I - where I is a unity operator.
| ih | | ih |
tobad-cavityregime. InSec.3wewillwritemasterequa- The fˆ (t), fˆ (t) and fˆ (t) are the Langevin noise
a σ D
tionforournanolaserinthecoherentstaterepresentation operators which arise through the interaction with the
andinstationaryregimederiveapproximatedexpression heat baths. Using the standard quantum-optical meth-
for Glauber-Sudarshan P function. With the help of the ods(see,forexample,Refs.[14]),weobtainthefollowing
2
nonzero correlation functions of these operators: where r = c/2 1 is the point where the stationary
m
−
solution has a maximum I =I (c/8 1).
fˆ(t)fˆ†(t′) = κδ(t t′), m s −
h a a i − The stationary intensity I0(r) increases as a pump
hfˆσ†(t)fˆσ(t′)i = Γδ(t−t′), is increased and it starts from the threshold point rth.
hfˆσ(t)fˆσ†(t′)i = γδ(t−t′), (2) When c ≫ 8 then rth ≈ 1+4/c, what indicates on fi-
fˆ (t)fˆ (t′) = 2γ σˆ(t) δ(t t′), nite value ofthe thresholdor,in other words,there is no
h σ D i h i − thresholdlessregimeinthissemiclassicalmodel. Theself-
hfˆD(t)fˆσ(t′)i = −2Γhσˆ(t)iδ(t−t′), quenching point rq corresponds to pumping rate where
fˆ†(t)fˆ (t′) = 2Γ σˆ†(t) δ(t t′), the atomic polarizationis rapidly dampedto zerodue to
h σ D i − h i −
emitter trapping in the excited state, what leads to the
fˆ (t)fˆ†(t′) = 2γ σˆ†(t) δ(t t′),
h D σ i h i − damping of the field.
fˆ (t)fˆ (t′) = 2[(Γ+γ) (Γ γ) Dˆ(t) ]δ(t t′).
D D FromtheexpressionforthemaximumofintensityI it
h i − − h i − m
follows that large mean number of photons in the cavity
We are also interested in the field properties transmit-
can be achievedwhen cI 8 (we assumed that c 8).
ted outside the cavity through the outcoupling mirror. s
≫ ≫
Thelatter conditioncanbe writteninotherformg κ.
For that we should consider the annihilation operator
≫
Thus the semiclassicalregime takes place when the pho-
aˆout(t) of photons outside the cavity [15]. The input-
tons lifetime in the cavity is so long that each photon
output transformation is
is provided by coherent interaction with the single emit-
aˆout(t)=√κaˆ(t)−fˆa(t)/√κ. (3) ter (above mentioned condition). The condition cIs ≫8
alsogivesuspossibilitytocollectalargemeannumberof
In the next Sec. 2.2 we will applied linearization pro-
photons in two opposite cases, namely in good - I 1
ceduretoEq.(1)whichallowsustoinvestigatethesmall s ≫
and bad-cavity regimes I 1.
fluctuations of laser field near the dominant stationary s ≪
semiclassical mean value. To find the latter for our
nanolaser we need to collect a large number of photons
inthe cavityandalsoprovidethe coherentinteractionof 2.2 Linearization around the stationary solution
each photon with the single emitter.
In the assumption that above two conditions are sat-
To linearizethe Heisenberg-Langevinequations(1) we
isfied let us obtain the stationary semiclassical solu-
assume that all operators can be presented as a sum of
tion from Eq. (1). For that we need to drop all time
derivatives dXˆ/dt and average the remained stationary udeodmflinuacnttuactlaiosnsicδaXˆl t(esreme,Xfo0r aexnadmap"les,mRaellf"s.o[1p6e,ra1t7o]r),val-
equations under assumption of following decorrelations
Dˆaˆ = Dˆ haˆi, σˆ†aˆ = σˆ† haˆi. Suppose that in Xˆ =X0+δXˆ. (6)
DsteadEy staDte Ethe op(cid:10)tical(cid:11)phas(cid:10)e is(cid:11) randomly distributed
between 0 and 2π we take the average value of opera- After linearizationwith respect to δXˆ we obtain the fol-
tors with the fixed arbitrary mean value of the phase to lowing equations for the fluctuations
be equal to zero ϕ0 = 0. Such selection of the phase
value implies the reality of semiclassical field amplitude d κ
δaˆ(t)= δaˆ(t)+gδσˆ(t)+fˆ(t),
a0 and polarization σ0 = 2κga0. Thus, the analytical ex- dt − 2 a
pressionfor semiclassicalstationaryintracavityintensity d 1
I0 = a0 2 is therefore dtδσˆ(t)= − 2(Γ+γ)δσˆ(t)+g a0δDˆ(t)+D0δaˆ(t)
| |
Is (r+1)2 + fˆσ(t), h i
I0 = 2 (cid:20)(r−1)− c (cid:21). (4) dδDˆ(t)= (Γ+γ)δDˆ(t) 2g σ0(δaˆ(t)+δaˆ†(t))
dt − −
where we introduce new dimensionless parameters: the h
dimensionless pumping rate r = Γ/γ; the dimensionless + a0(δσˆ(t)+δσˆ†(t) +fˆD(t). (7)
saturationintensityIs =γ/κ;thedimensionlesscoupling i
strength c=4g2/κγ. BeforesolvingEqs.(7)wefirstsplitoperatorsintoHer-
Eq. (4) coincides with one firstly obtained by Mu and mitian "real" and "imaginary" parts as
Savage [1]. I0(r) is a parabolic function of pump rate
which has physical interpretation when c > 8 and when
δaˆ(t) = δuˆ(t)+iδνˆ(t),
the value of the pump rate lying in the domain between
δσˆ(t) = δµˆ(t)+iδηˆ(t),
two points, so called threshold r and self-quenching r
th q
points, which are given by the following expressions fˆ(t) = Σˆ(t)+i∆ˆ(t). (8)
c c
rth =rm 1 8/c, rq =rm+ 1 8/c, (5) In this way the linearized equations of motion separate
− 2 − 2 −
p p
3
into two independent blocks "imaginary" parts is a result of our phase selection in
the derivation of semiclassical solution (see Eq. (4) and
d κ
δuˆ(t)= δuˆ(t)+gδµˆ(t)+Σˆ (t), text above).
a
dt − 2 The block for three real parts δuˆ, δµˆ, δDˆ are related
d 1
δµˆ(t)= (Γ+γ)δµˆ(t)+g a0δDˆ(t)+D0δuˆ(t) to the intensity fluctuations via δIˆ = 2a0δuˆ. The two
dt − 2
imaginarypartsδνˆ,δηˆcanbeassociatedwithphasefluc-
+ Σˆ (t), h i
σ tuations.
d
δDˆ(t)= (Γ+γ)δDˆ(t) 4g σ0δuˆ(t)+a0δµˆ(t) Eqs.(9,10)canberesolvedbymeansofFouriertrans-
dt − − formation. For that we need to perform the fluctuations
+ fˆD(t); h i(9) as
∞ dΩ
dδνˆ(t) = κδνˆ(t)+gδηˆ(t)+∆ˆa(t), δXˆ(t)=ˆ δXˆ(Ω)exp(−iΩt)2π, (11)
dt −2 −∞
d 1
dtδηˆ(t) = −2(Γ+γ)δηˆ(t)+gD0δνˆ(t)+∆ˆσ(t). which gives us the linear algebraic equations which can
(10) beresolvedbymeansofCramer’srule. Theresultforthe
Fourier-transformedamplitude δuˆ and phase δνˆ quadra-
We want to note that the independence of "real" and ture components are
Σˆ (Ω)A(Ω)+Σˆ (Ω)C(Ω)+fˆ (Ω)B
a σ D
δuˆ(Ω) = , (12)
iΩ (iΩ (Γ+γ+κ))/2)(iΩ (Γ+γ))+4g2 a0 2 4κg2 a0 2
− − | | − | |
g∆ˆh (Ω) ∆ˆ (Ω)(iΩ (Γ+γ)/2) i
σ a
δνˆ(Ω) = − − , (13)
iΩ iΩ (Γ+γ+κ)/2
−
h i
where Eqs. (2) and from Eq. (11) the Fourier-transformed
quadraturecomponents δuˆ, δνˆ are δ-function correlated.
A(Ω) = [(iΩ (Γ+γ)/2)(iΩ (Γ+γ))+4g2 a0 2], Using the input-output transformation Eq. (3) we find
− − − | |
C(Ω) = g(iΩ (Γ+γ)), B = g2a0. (14) the amplitude and phase fluctuation spectra
− −
As it follows from the correlation function definition
1 κ2 aΩ2+bγ2
hδuˆout(Ω)δuˆout(Ω′)i= 4δ(Ω+Ω′) 1+ 4 Ω2(Ω2 d)2+γ2(eΩ2 f)2 , (15)
h − − i
1 κ2 γ2[c(r+1)+(r+1)2]
hδνˆout(Ω)δνˆout(Ω′)i= 4δ(Ω+Ω′) 1+ 4 Ω2(Ω2+γ2(r+1+1/I )2/4) , (16)
s
h i
where Now we will use these results to analyze two different
regimes of our nanolaser, namely good- and bad-cavity
regimes.
a = γ2[c(r 1) (r+1)2+2cr],
− −
b = γ2(r 1)[3(r+1)3+2c2 c(r+1)(r 5)],
− − −
cI (r 1)+(r+1) 3(r+1)+1/I
d = γ2 s − , e= s,
2I 2
s
f = γ2cI0/Is2. (17)
4
2.3 Good-cavity regime Another quantity of interest is the low-frequency
asymptotic version of the fluctuation spectrum of the
In the case of good-cavity regime I 1 the ob- phase quadraturecomponent. Using the Eq. (16) we ob-
s
tainedexpressionfortheamplitudefluctuat≫ionspectrum tain
Eq. (15) becomes κ21+c/(r+1)
hδνˆout(Ω)δνˆout(Ω′)i=δ(Ω+Ω′) 4 Ω2 . (22)
1 S(r,c)
hδuˆout(Ω)δuˆout(Ω′)i= 4δ(Ω+Ω′) 1+ 1+(Ω/Ω0)2 . The low-frequency divergence in this spectrum as 1/Ω2
h (1i8) is manifesting for the phase diffusion process [18]. This
The first term in the square brackets corresponds to spectrum also defines the linewidth ∆ν of our nanolaser
standard quantum limit (SQL), the second term is a through (δνˆout)2 Ω/I0
h i
Lorentzian with width Ω0 given by
1 c
(r+1)2 ∆ν =∆νST2 1+ r+1 , (23)
Ω0 =κ 1 (19) (cid:18) (cid:19)
− c(r 1)
h − i where∆νST =κ/(2I0)istheSchawlow-Towneslinewidth
of a conventional incoherently pumped laser. The ob-
and this term describes the additionally exes above the
tainedformulaforthelinewidthshowsthedeviationfrom
SQL or reduction of the noise below SQL depending of
the Schawlow-Townes result and this coincides with the
the sign of the strength S(r,c)
numerical simulations performed by Clemens et al [9].
(r 1) 3(r+1)3+2c2 c(r+1)(r 5) At the end of this section we want to discuss the le-
S(r,c)= − − − .
[c(r 1) (r+1)2]2 gality of above linearizationprocedure. Let us explain it
(cid:2) (cid:3)
− − interms of adiabaticeliminationofthe emitter variables
(20)
in Eq. (1). Indeed, in the case when I 1 one can ob-
We have investigated this expression in order to find s
≫
tain the Heisenberg-Langevinequation only for the field
outa possibility of the noisereductionbelow the SQL in
operator aˆ(t). The semiclassical solution of the latter
thelaserlightoutsidethecavity. Unfortunately,thiskind
coincides with Eq. (4). The following linearization with
of nonclassical phenomenon are very limited. We have
respect to fluctuation δaˆ(t) gives us equation which con-
foundthatinthe regionofpumpparameterr <r <r
m q
thestrengthS(r,c)isalwayspositiveandincreasingfunc- tains the products like δaˆ(t)fˆα(t) (α = σ, D). Because
tion, resulting in the excess noise in Eq. (18). The small ofpresencejust oneemitter they cannotbe neglectedin
negative values of S(r,c) are observed when the pump frame of linearization procedure. However, these prod-
parameter is centered around r = c/5 (c 200), i.e. ucts appear in the equationas terms multiplied by small
lviaelsuienstahreertehgeiorensurtlhto≪ftrhe<wremll.-kPnroowbnabalnyt,ibthu≥enscehnineggapthivee- ftahcetoprro2dκu/c(tΓs−simγi)la<rt1o/Is0ec≪ond1,owrdheartinmflakuecstutahteioonrdδeaˆr(to)f.
nomenon for a single-emitter. However, this antibunch- Finally, the fully linearized equation for aˆ(t) gives the
ing effect is strongly attenuated due to the effect of ac- same results as was obtained in this section.
cumulationofmanyphotonsinside the cavityduringthe
long (and random) photon lifetime.
2.4 Bad-cavity regime
The Mandel Q-parameter can be expressed through
the amplitude quadrature component δuˆ as Q = 4
∞ δuˆ(Ω)δuˆ( Ω) dΩ/2π 1 and the result is · The considered single-emitter laser relates to so called
´−∞h − i − high β lasers - the lasers with high fraction β of spon-
1 (r+1)2 taneous emission into the lasing mode (see expression
Q(r,c)= S(r,c) 1 . (21)
for β in Sec. 3, Eq. (44)). Most of present work on
2 − c(r 1)
(cid:20) − (cid:21) the intensity noise in high β lasers has been applied to
We see that the strength S(r,c) is also characterized standard semiconductor laser diodes, embedding quan-
this quantity. On the Fig. 1a there are some graphics tum wells as gain material. These models predict that
for Mandel Q-parameter for different coupling constant if β is small, the peak-to-valley difference of the relax-
c when I 1. Forthe pump rate lying betweenthresh- ation oscillations is very large and it decreases with in-
s
≫
old and quenching points our theoretical results are in creasing β [19, 20]. However, all these models are based
a very good agreement with the exact solution of the on standard rate equations used in quantum well lasers
master equationfor the density matrix (see next Sec. 3). and very few studies have been carried out in this field
The main divergency appears near the threshold and whenswitchingfromquantumwelllaserstoquantumdot
quenching points where the linear theory does not work. lasers. Some recent research was focused on the turn-on
Two fluctuations peaks of Mandel Q-parameter are well dynamicsandrelaxationoscillationinstandardquantum
known[5]: firstofthemassociatedwithlaserturnonand dots lasers [21] but has not been extended to unconven-
second broad peak associated with laser turn off. tional high β nanolasers.
5
authors. Thus in [1] the master equation have been
reduced to system of first order, ordinary differential
equations and analyzed numerically. In [5, 9] more ef-
ficient quantum trajectory algorithm have been devel-
oped for numerical evaluating of master equation. The
works [6, 7] are devoted to analysis of the master equa-
tion written in terms of coherent states for field. In
light of our interest, we want to briefly discuss one of
them, namely [6]. Authors have worked with the sys-
tem of equations for Glauber-Sudarshan P function and
additionalquasi-probabilities. Besidesnumericalsimula-
tions they have also found approximated expression for
P function which can be used in the case of good-cavity
regime. Here we follow the nearest approach as in [6],
butwe manageto deriveisolatedstationaryequationfor
P function. A detailed analysis of the latter allows us to
obtain approximatedsolution, which has wider regionof
application than that in [6].
3.1 Coherent state representation of master equation
The master equation for our nanolaser is (see, for ex-
ample, [9])
Figure 1: a) The Mandel Q-parameter (21) vs the pump pa- ∂ρˆ i κ
rameter r. Dashed line shows results obtained with help of ∂t = − ~ Vˆ,ρˆ + 2 2 aˆρˆaˆ†−aˆ†aˆρˆ−ρˆaˆ†aˆ
fcuunrvcteisonIsP=0 (2306.).b)1T)hce=sp2e0c;tr2u)mco=f a3m0;pl3i)tucde=fl4u0c.tuFatoironasll. + γh2σˆρˆiσˆ† σˆ(cid:0)†σˆρˆ ρˆσˆ†σˆ (cid:1)
2 − −
1) r=rm =199, 2) r=3. Forall curvesIs=0.2, c=400.
Γ(cid:0) (cid:1)
+ 2σˆ†ρˆσˆ σˆσˆ†ρˆ ρˆσˆσˆ† , (24)
2 − −
Here we use results obtainedinSec. 2.2 to analyzebe- (cid:0) (cid:1)
where interaction between the cavity mode and the sin-
haviourofournanolaserinthe caseofbad-cavityregime
gle two-level system is given by the Jaynes-Cummings
I 1. As was mentioned above, for that we only need
s ≪ Hamiltonian
to preserve the following condition cI 8.
s
≫
If the pump rate r is close to the maximum point rm Vˆ =i~g aˆ†σˆ σˆ†aˆ . (25)
the amplitude fluctuations spectrum is described by the −
sameEqs.(18,22)asinagood-cavityregime(seeFig.1b, (cid:0) (cid:1)
Finally, we want to obtain equation for P function, for
curve 1). The main difference is the magnitude of the
that we need to rewrite Eq. (24) in terms of coherent
width Ω0 (19) for the amplitude fluctuations spectrum. states for field z , z∗ and in projections of the density
Other situation takes place when the pump rate r lies | i | i
operatoronthetwo-levelsystemstates 1 and 2 . Using
notfarfromthethresholdpointr . Inthiscasethehigh | i | i
th well known rules (see, for example, Ref. [22])
peakintheamplitudefluctuationsspectrumappears(see
Fig. 1b, curve 2). The physical origin of this peak is the aˆρˆ zρˆ(z,z∗),
relaxation oscillation phenomenon. Because of lineariza- →
∂
tion procedure break down in the vicinity of threshold ρˆaˆ z ρˆ(z,z∗),
→ − ∂z∗
rth, we can not correctly define width and height of this (cid:20) (cid:21)
peak,butwecanestimatethefrequencyoftherelaxation ρˆaˆ† z∗ρˆ(z,z∗),
→
oscillations Ω /κ=I c(r 1)/2. ∂
osc s − aˆ†ρˆ z∗ ρˆ(z,z∗), (26)
→ − ∂z
p (cid:20) (cid:21)
3. MASTER EQUATION APPROACH where z, z∗ are the complex variables and intro-
duce the following quasi-probabilities ρ (z,z∗) =
ik
This section are devoted to investigation of the i ρˆ(z,z∗) k ρik (i,k = 1,2), D = ρ22(z,z∗)
h | | i ≡ −
nanolaser properties with help of master equation for ρ11(z,z∗) and P function P = ρ11(z,z∗) + ρ22(z,z∗),
density matrix. This approach have been used by many weobtainfromEq.(24)the systemofpartialdifferential
6
∗
equations gives the following relations ρ21 = (ρ12) = (κ/2g)zP.
The latter allows us to eliminate ρ21, ρ12 from the sta-
∂P ∂ κ ∂ κ
= zP gρ21 + z∗P gρ12 , tionary form of the system (27) and obtain two coupled
∂t ∂z 2 − ∂z∗ 2 − 2
equationsforP andD. InthenewvariablesI = z and
∂D (cid:16) (cid:17) ∂(cid:16) κ (cid:17) | |
= (Γ γ)P (Γ+γ)D+ zD+gρ21 ϕ (z =√Iexp(iϕ)) these equations read
∂t − − ∂z 2
∂ κ (cid:16) (cid:17)
+ ∂z∗ 2z∗D+gρ12 −2g[z∗ρ21+zρ12], IP Is [(r 1)P (r+1)D]= 1 ∂ I(P +D), (28)
− 2 − − 2∂I
∂∂ρt21 = −(Γ(cid:16)+2 γ)ρ21+ ∂∂z (cid:17)κ2zρ21 + ∂∂z∗ κ2z∗ρ21 IP(r+1+1/Is) ID+ 1(P +D)
c − 2
1 ∂ (cid:16) (cid:17) (cid:16) (cid:17) (cid:20) (cid:21)
+ g zD (P +D) . (27) 1 ∂ 4 i ∂
− 2∂z∗ = I2P I(P +D) (P +D),(29)
(cid:20) (cid:21) 2∂I cI − − 4∂ϕ
(cid:20) s (cid:21)
The first equation in this system is written in special
form which can be associated with conservation law for where we introduced above mentioned dimensionless pa-
the quasi-probability ∂P/∂t = divJ, where J is known rameters I , c, r.
s
as probability current density. The additional quasi- There is no phasing influence on our nanolaser, e.g.
probabilities D and ρ21 have a clear physical meaning. - external field with fixed phase or coherent pumping.
Thus, the mean values for inversion and polarization of Therefore expected stationary solution does not depend
the two-levelsystemareexpressedthroughD andρ21 as on phase and we can write P and D as only function
Dˆ = D(z,z∗)d2z, σˆ = ρ21(z,z∗)d2z. of I = z 2. This is also following from the second
h i | |
´ ´ Eq.(29): thetermproportionaltoimaginaryunitshould
D E
be equated to zero because of reality of functions P and
3.2 Stationary solution D.
Let us derive isolated equation for the P function in
In stationary regime the probability current density J thestationaryregime. UsingEqs.(28,29)wecanexpress
canbeequatedwithzero(see,forexample,[23,24])what function D in terms of P
1 4 ∂ 2[3 I (r+1+c)]
D = I2 P + − s IP +(r 1+1/I )P . (30)
(r+1 1/I ) 2I/I cI2 ∂I cI2 − s
− s − s (cid:18) s s (cid:19)
Nowputthisrelationintothefirstorthesecondequation In the case of good-cavity regime functions p(I) and
inthesystemEqs.(28,29). Insuchawaywegetisolated q(I) in Eq. (31) contain large parameters like cI , I3 or
s s
equation for P function I2(seeAppendix). Therefore,wecantrytofindapproxi-
s
matedsolutionofEq.(31)usingtheperturbationmethod
P′′(I)+p(I)P′(I)+q(I)P (I)=0, (see, for example, Ref. [25]). The point of this method
p(I)= a10+a11I +a12I2 / a02I2+a03I3 , is representation of the unknown function P as iterative
series P = ∞ λ−nP , where λ is a large parameter.
q(I)=(cid:0)a20+a21I+a22I2(cid:1)/(cid:0)a02I2+a03I3(cid:1), (31) n=0 n
AsubstitutionofthelatterseriesintoEq.(31)withsub-
P
(cid:0) (cid:1) (cid:0) (cid:1) sequent equating to zero of terms with same powers 1/λ
wheretheprimeindicatesderivativewithrespecttovari-
2 gives possibility to find the different orders of approxi-
ableI = z andfunctions p(I),q(I)dependonphysical
| | mations P .
parameters I , c, r via a (see Appendix). n
s ik
To isolate the correct single large parameter let us
The obtained equation is the second order differential
rewrite p(I) and q(I) in the following form
equation with polynomial coefficients. To use this equa-
tion we need to define boundary P(0), P( ) or "ini-
tial" P (0), P′(0) conditions. If we equate t∞he variable p(I)=a12(I −I−1)(I−I+1)/I2(I−I00), (32)
I to zero in Eq. (31) we obtain the following relation q(I)=a22(I I−2)(I I+2)/I2(I I00), (33)
− − −
P′(0)= a20/a10P (0). This relation together with nor-
malizatio−nofthequasi-probabilityP givesacomfortable where roots are
way to define the "initial" condition for numerical simu-
2
lations. I±i = ai1/2ai2 (ai1/ai2) 4ai0/ai2/2, I00 = a02.
− ± − −
q
7
In the conditions I 1, c > 8 the magnitudes of the rate r can be comparable or bigger than c (for example
s
≫
roots I+1, I+2, I00 approximately equal each other and near to maximum rm or to quenching rq points).
therootsI−1,I−2 havethesimilarorder. So,itisnatural Thus, in zero-order approximation (first term in the
to save all I±i and extract the large parameter from the seriesP = ∞ λ−nP )weneglecttheterminEq.(34)
constants a12, a22 in Eqs. (32, 33). This parameter is proportional nto=01/λ whnat implies first-order differential
therefore λ = cIs(=a22 ≈−a12) and Eq. (31) can be equation foPr function P0
written as follows
λ−1P′′(I)+p˜(I)P′(I)+q˜(I)P (I)=0, (34) a12(I I−1)(I I+1)P0′(I)
− −
where p˜(I) = p(I)/a22, q˜(I) = q(I)/a22. Here we want + a22(I−I−2)(I −I+2)P0(I)=0. (35)
to note that we do not neglect the term r/c in relation
a12/a22 =(7 3Is(r+1) 2cIs)/2cIs,becauseofpump P0 is then found from Eq. (35):
− −
P0(I)= N0(1−I/I−1)f1(1−I/I+1)f2exp −aa1222I , if I ≤I−1 (36)
( 0, (cid:16) (cid:17) otherwise
a22(I−1 I−2)(I−1 I+2) a22(I+1 I−2)(I+1 I+2)
f1 = − − , f2 = − − ,
−a12 (I−1 I+1) a12 (I−1 I+1)
− −
where N0 is normalizing constant. The roots and con- the equation (31) (solid gray line). The different curves
sequently the powers f1, f2 in Eq. (36) have complex correspond to different values of pump rate r while c
structure and will be estimated below in some special and I are fixed. We see a very good agreement with
s
cases. numerical results.
To understand why our solution is restricted we note The main divergency appears for value of pump rate
thatthe rootsI±1 offunction p(I)inEq.(31)areknown discovered in Sec. 2.3, namely for r = c/5, where the
as turning points, where oscillating behaviour of P (I) Mandel Q-parameter Q(r,c) (21) has a minimum (see
can change on exponential. A careful examination of Eq.(20)andtextbelow). IntherightwingofPfunction
Eq. (31) shows that in the conditions Is 1, c > 8 on (Fig. 2b) the small oscillations appear when c becomes
≫
the I-domain ranging from 0 to turning point I−1 the comparable with Is, but still our solution is good. With
P (I) is positive and nonoscillating function. Moreover, growing of c the oscillations become more pronounced
its essential changing occurs just on latter domain (it and the maximum of P function goes out of domain
is not for every set of parameters r, Is, c, see below). (0,I−1) and our solution (36) becomes inapplicable in
Otherwise, when I >I−1 then P (I) becomes oscillating vicinity of r = c/5. It is interesting to note that critical
function. TheobtainedsolutionEq.(36)cannotdescribe situationoccurswhen c>200. Forlattervalues ofcand
any oscillations, moreover it possesses imaginary values for arbitrary saturation intensity I the maximum of P
s
when I > I−1, therefore we have restricted definition function goes out of the domain (0,I−1). As mentioned
domain for our solution by the value I−1 and call them above, the P function acquires oscillating character for
boundary root. I >I−1, what probably indicates on the nonclassical ef-
To reveal physical meaning of the boundary root we fect associated with photon antibunching.
have simplified expression for I−1 and found maximum Now let us analyze the obtained solution (36) in some
of function P0(I) on the domain (0,I−1). When Is 1, special cases.
≫
c > 8 we have managed to estimate the latter quan-
tities with help of results obtained in Sec. 2.3: I−1
≈
Isr/(2+3r/c) I0[1+Q(r,c)] and the point where
P0(I) reaches h≥is maximum is I = I−2 I0. Thus, Far below threshold and far above quenching
≈
Pfunctionhasamaximumatthepointcorrespondingto
the semiclassical intensity I0 Eq. (4) and the boundary At first we consider situation when the pump rate lies
root is at a distance I0Q(r,c) from it, where Q(r,c) is far below threshold r rth. In this case the average
the Mandel Q-parameter Eq. (21). value of I = z 2 is sm≪all and the following inequalities
| |
In the Fig. 2 we plot the P function calculated using I/I−1 I/(Isr/2) 1, I/I+1 I/(Is(r+1)/2) 1
≈ ≪ ≈ ≪
Eq.(36)(dashedline)andusingnumericalsimulationsof are satisfied. Thus, the factor in front of exponential in
8
function P0(I)=δ(I), what coincides with the thermal
distribution with zero temperature.
Near to maximum point rm
Here we consider situation when the pump rate
is in the vicinity of r , where the semiclassical so-
m
lution I0 Eq. (4) reaches his maximum. In this
case the powers in Eq. (36) are simplified as f1
≈
I 3(r+1)3+2c2 c(r+1)(r 5) /(3(r+1)+2c)2,
s
− −
f2(cid:16) 0 and the root I−2 becomes sim(cid:17)ilar to I0. Simple
≈
algebra leads to following expression for P0
α
P0(∆I)= N0 (1−∆I/Λ)Λexp(∆I) , if ∆I 6Λ
( h 0, i otherwise
(39)
where we introduced new variable ∆I =I I0 and con-
−
stants Λ = f1/α, α = 2c/(3(r+1)+2c). For chosen
valuesofpump ratethe averagevalue of∆I is smalland
the inequality∆I/Λ 1is satisfied. Thus forthe factor
≪
in front of exponential in Eq. (39) we can write
Figure2: a)PfunctionvsI =|z|2 fordifferentpumpparam- (1 ∆I/Λ)Λ = exp[Λln(1 ∆I/Λ)]
− −
eter r. 1) r = 8, 2) r = 16, 3) r = 24, 4) r = 36, 5) r = 44. exp ∆I ∆I2/2Λ , (40)
For all curves Is = 100, c = 50; b) Oscillation behaviour of ≈ − −
the P function, Is = c = 100, r = c/5 = 20. Dashed line - what implies Gaussian P func(cid:2)tion (cid:3)
solution (36), solid gray line - numerical simulation, dotted
line - solution from Ref. [6]. 2
(I I0)
P0(I)=N0exp − . (41)
"− 2Λ/α #
Eq. (36) can be approximated as follows
Here we removed the restriction on P0. Indeed, the dis-
(1 I/I−1)f1(1 I/I+1)f2 tribution (41) never really sees the boundary and I can
− − be taken to run from 0 to .
=exp[f1ln(1 I/I−1)]exp[f2ln(1 I/I+1)] ∞
− − Using the obtained Eq. (41) we can estimate the av-
exp[ (f1/I−1+f2/I+1)I]. (37) erage value of photons in the cavity nˆ and its variance
≈ −
2 h i
(∆nˆ)
The P0 is therefore h i
∞
P0(I)= (a10/1a20)exp(cid:18)−(a10I/a20)(cid:19). (38) hnˆi = ˆ0 IP0∞(I)dI ≈I0, (42)
2 2
(∆nˆ) = nˆ + (I nˆ ) P0(I)dI
The obtained expression for P function corresponds to h i h i ˆ −h i
0
thermal distribution with the following mean number of I0[1+Λ/(αI0)]. (43)
photons in the cavity nˆ =a10/a20 r. ≈
h i ∼
For the values of pump rate lying far above quench- A substitution of the explicit form of Λ and α into the
2
ing point r rq we can make the same manipulation expression for variance gives (∆nˆ) = I0[1+Q(r,c)],
≫ h i
with Eq. (36) as above. Using inequalities I/I−1 where Q(r,c) is the Mandel Q-parameter obtained from
≈
I/(Isr/(2+3r/c)) 1, I/I+1 I/(Isr/2) 1 which linear theory (21).
≪ ≈ ≪
occur in this case, we get the same thermal distribution In the weak coupling regime c < I the obtained
s
as Eq. (38), but the mean number of photons possesses Eq. (41) works good for values of pump rate lying in
another behaviour nˆ =a10/a20 1/r. whole semiclassical region rth < r < rq. In the strong
h i ∼
In the two limiting situations, when r 0 or r coupling regime c >I the symmetrical Gaussian distri-
s
→ →
, the correspondingmean number of photons converge bution (41) is not sufficient and it becomes more pro-
∞
to zero and the distribution Eq. (38) gives Dirac delta- nounced near to mentioned value of pump rate r =c/5.
9
Threshold
In our previous report [13] we considered nanolaser
behaviour around the semiclassical threshold r (5)
th
and specified the condition required for thresholdless
regime. Herewecontinueourresearchusingtheobtained
Eq. (36).
The behaviourofPfunctioninthe thresholdpoint r
th
forthree differentregimes1)c I , 2)c I , 3)c I
s s s
≪ ≈ ≫
is shown in Fig. 3a. To realize these regimes we fix the
large coupling constant c = 100 and set the different
values of the saturation intensity: I = 600, 60, 6 (we
s
choose such values of I for better resolution of P func-
s
tion behaviour). In the weak coupling regime (c I )
s
≪
the P function has a typical plateau (it is marked by
solid straight line, curve 1), which indicates transition
to lasing: from thermal to Gaussian type distribution.
As saturation intensity is decreased and the strong cou-
plingregime(c I )occursthemaximumofPfunction
s
≫
movesfromzerovalue ofvariableI andthe semiclassical
threshold behaviour disappears (curve 3).
Above we obtained the maximum point for P0 (36)
and it was the root I−2. To define the threshold value
of pump rate we need to solve equation I−2(r) = 0, i.e.
find such value of r when the maximum of the P func-
tion is in the point I = I−2(r) = 0. The approximate Figure3: Pfunction vsI =|z|2. a) Transition tothethresh-
solution is r˜ = 1+4/c 2/I . When c I then the
− s ≪ s oldless regime. 1) Is = 600, 2) Is = 60, 3) Is = 6. For all
last term can be neglected and r˜ equal to semiclassical
curves c = 100, r = rth = 1.04. b) Parameters was taken
threshold rth ≈ 1 +4/c > 1 (5). When c ≫ Is then as in Ref. [6]: 1) c = 2000, Is = 5, r = 6, 2) c = 1428.57,
r˜ becomes smaller than unity what indicates the disap- Is = 7, r = 4.28, 3) c = 1000, Is = 10, r = 3, 4) c = 500,
pearance of the semiclassical threshold and transition to Is=20, r=1.5, 5) c=285.7, Is=35, r=0.86. Dashed line
thresholdless regime (see the vanishing of narrow peak - solution (36), solid gray line - numerical simulation, dotted
in the behaviourofMandel Q parameterin ourpreviews line - solution from Ref. [6].
report [13], Fig. 2). The dynamics of r˜ in above three
considered regimes is r˜ 1.04, 1, 0.66 (remind that
r =1.04). ≈ formula for P function. The structure of their solution
th
has an identical form with our Eq. (36) (see formula (9)
At the end of this section we want to correct the mis-
in[6]),butthissolutionworksgoodinthe regionnearto
print occurred in our previous report (see formula (13)
semiclassical threshold r or quenching points r . For
in [13]). The valid expression for fraction β of sponta- th q
the pump rate lying around the maximum point r it
neous emission into the laser mode is m
gives uncorrect result for width of function P and as a
β =c/[(c+1)+I (r+1)]. (44) consequence the uncorrect variance of the photon distri-
s
bution.
We should note that this misprint did not affect on all In the Fig. 3b we plot some graphics of quasi-
our results and discussions. probabilityPforparameterstakenfrom[6]. Itisclearto
see that our Eq. (36) gives the same result as a solution
(9) obtained in [6] and all of them are in a good agree-
Comparing with other authors ment with numerical simulations. The Fig. 2a,b demon-
stratethatforthepumpratelyingaroundthemaximum
Inintroductiontothissectionwementionedthatclose point rm our Eq. (36) (dashed line) has an advantage
approach based on quasi-probabilities have been consid- with (9) [6] (dotted line). The latter indicates well the
ered by two authors Karlovich and Kilin in [6]. They maximumoftrueP function,but the widthisuncorrect.
have workedwith first orderdifferential equationsystem The derivationof a single equationforP function (31)
for function P and additionalquasi-probabilities. By ne- andaccurateextractionoflargeparametersgivesuspos-
glectingofsmallparametersinthissystemtheyhaveob- sibility to obtainedthe solutionwhich has more wide re-
tainedintegrableequationsandasaresultthe analytical gion of application.
10
4. SUMMARY ACKNOWLEDGMENTS
This work was supported by French National Agency
In this paper we have studied physical properties of a
(ANR) through Nanoscience and Nanotechnology Pro-
single-emitterlaser. Theproblemhavebeeninvestigated
gram(ProjectNATIFn◦ANR-09-NANO-012-01),bythe
in terms of the Heisenberg-Langevin equations and in
CNRS-RFBR collaboration (CNRS 6054 and RFBR 12-
termsofthemasterequationfordensitymatrix. Inboth
02-91056) and by external fellowship of the Russian
approacheswehaveprovidedanalyticalresultswhichare
Quantum Center (Ref. number 86).
summarized below.
In the case of good-cavity regime with help of the
Heisenberg-Langevinapproachwe have obtainedanalyt-
APPENDIX
ical expressions for linewidth Eq. (23), amplitude fluc-
tuation spectrum Eq. (18) and Mandel Q parameter
The constants a in Eq. (31)
ik
Eq. (21). These results work good in the semiclassical
region of pumping rate r < r < r . According to
th q 1 (r+1)
nanolaserbehaviourthelatterregioncanbesplitintotwo a02 = Is ,
2 − 2
subregions. In the first subregion r < r the nanolaser
m a03 = 1,
behaviour is similar to that of conventional laser. Also
cr cr(r+1)
in this subregion for r centered around r = c/5 and for a10 = Is2 4 −Is3 4 ,
largecoupling constant(c>200)we havediscoveredthe
9 6(r+1)+c 3(r+1)2+c(4r+2)
small noise reduction below the SQL in the laser light a11 = 4 −Is 2 +Is2 4 ,
outside the cavity and small negative values in Q(r,c)
7 3(r+1)+2c
(21). The master equation approach confirms this non- a12 = Is ,
classical phenomenon: P function manifests oscillating 2 − 2
bofehthaeviwouelrl.-kPnroowbnabalnyt,itbhuenscehnineggapthiveenovamluenesonarfeorthaesrinesgulelt- a20 = 32 −Is11(r+41)+3c +Is23(r+12)2+2c
emitter. Inthe secondsubregionr>r the excessnoise c(r+1) (r+1)2
m + I3 (r 1) ,
is always takes place. The P function in this subregion s 4 − − c
(cid:20) (cid:21)
does not have any nonclassical features. 3 2cr (r+1)2
In the bad-cavity regime we have observed two differ- a21 = 2 −2Is(r+1)−Is2 − 2 ,
entsituations: whenthepumprater isclosetoitsmaxi- a22 = cIs. (45)
mum value r then lasergeneratesasin the good-cavity
m
regime;whenrisclosetothresholdr thenhighpeakin
th
the amplitude fluctuationsspectrumappearswhichindi-
cates on the relaxation oscillation phenomenon.
Withthehelpofthemasterequationfordensitymatrix ∗ Electronic address: [email protected]
written in terms of coherent states we have managed to [1] Yi Mu, C. M. Savage, Phys.Rev.A. 46, 9, 5944 (1992).
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[8] S. Ya. Kilin and A. B. Mikhalychev, Phys. Rev. A 85,
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expressionforthresholdpumprater˜=1+4/c 2/I . In [9] J. P. Clemens, P. R. Rice, and L. M. Pedrotti,
s
− J. Opt.Soc. Am. B 21, 2025 (2004).
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s
≫ [10] J. McKeever, A. Boca, A. D. Boozer, J. R. Buck, and
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H. J. Kimble, Nature425, 268 (2003).
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th s [11] F. Dubin, C. Russo, H. G. Barros A. Stute, C. Becher,
≈ ≪
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