Transient dynamics of linear quantum amplifiers

0.26 MB·

by  S. Maniscalco

EPJ manuscript No.
(will be inserted by the editor)
Transient dynamics of linear quantum amplifiers
Sabrina Maniscalco1, Jyrki Piilo1, Nikolay Vitanov2, and Stig Stenholm3
6 1 School of Pureand Applied Physics, University of KwaZulu-Natal, Durban 4041, South Africa
0 2 Department of Physics, SofiaUniversity,James Boucher 5 Boulevard, 1164 Sofia, Bulgaria
0 3 Laser Physics and Quantum Optics, Royal Institute of Technology (KTH), Alba Nova, Roslagstullsbacken 21, SE-10691
2
Stockholm, Sweden

n
a Received: date/ Revised version: date
J

3
Abstract. The transient dynamics of a quantum linear amplifier during the transition from damping to
1
amplification regimeisstudied.Themasterequationforthequantizedmodeofthefieldissolved,andthe

solution is used to describe the statistics of the output field. The conditions under which a nonclassical
2
input field may retain nonclassical features at the output of the amplifier are analyzed and compared to
v
8 the results of earlier theories. As an application we give a dynamical description of the departure of the
1 system from thermal equilibrium.
0
2 PACS. 42.50.Ar – 42.50.Dv – 42.50.Lc
1
4
0 1 Introduction plifierstothecaseinwhichasmoothonsetofamplification
/ or attenuation takes place, and hence the amplifying and
h
p The master equation describing linear amplification or attenuation coefficients are time dependent. We solve the
- gain has been a prototype for discussing many questions masterequationwithtimedependentcoefficientsinterms
t
n in Quantum Optics. It was derived by the elimination of ofthecharacteristicfunction[5,6]andweusethesolution
a anunobservedenvironmentusingwhathasbeentermeda to describe the transient dynamics of the linear quantum
u amplifier, one of the most widely used and common de-
Born-Markovapproximation[1]. Thus its properties were
q vices in quantum optics.
mainly determined by physical considerations, but it ar-
:
v rived at a form later to be shown to be the consistent
Wechoosetoconsiderasituationwherethegainmedium
Xi generator of dissipative time evolution in quantum the- is switched smoothly from an attenuating regime to an
ory; thus it is of the Lindblad form [2].
r amplifying one. This takes place within a time interval
a The master equation for linear amplification was ear-
centered at some definite time, before which we have a
lierrepresentedasthe genericmodelfor anopticalampli-
damped situation. Thus the initial gain, which is normal-
fier or attenuator [3]. It also describes the onset of laser
ized to unity of course, decreases first until the amplifi-
oscillationsuntilthetimewhennonlinearsaturationstarts
cation coefficient changes sign and gain starts to grow.
to affect the behavior. In the trapped ion context, when
The model allows us to follow the solution through this
the ion trap potential is regarded in a harmonic approx-
point, and we can see how the time dependence affects
imation, the cooling by lasers may be considered as an
the noise properties and the possibility to retain initially
attenuation described by the same equation [4].
imposed nonclassical features of the system state. As we
Itsadvantageisthatitisexactlysolvable,whichallows
may expect, the situation is more complicated than the
ustofollowtheonsetofgainorthedampedapproachtoa
simpleconstantcoefficientcase.Itis,however,possibleto
steadystate.Theexactsolutionalsoallowsustoevaluate
retain earlier results on amplifier added noise and quan-
the noise properties exactly and investigate the fading of
tumcloninglimitsbyconsideringtheappropriatelimiting
nonclassical features of the initial state.
cases.
In all applications so far, the amplifying and attenu-
ating coefficients of the equation have been regarded as Theoreticalworksonopticaltransientsofphysicalphe-
constants. This corresponds to the assumption that the nomena which cause amplification of light have received
population inversion is instantaneously reached and the inthe past a huge dealof attention[7]. To the best ofthe
evolution starts from an initial state experiencing no pre- authors’ knowledge, however, this is the first analytic de-
vious evolution. Such theory, however, does not describe scriptionofthetransientofphase-insensitivequantumlin-
the transient dynamics of the linear amplifier, i.e. the dy- ear amplifiers. Therefore the results presented in this pa-
namics when the pumping field is switched on or off. In per give a clear contribution to the fundamental research
this paperwe generalizethe previoustheory oflinear am- in the theory of lasers and optical amplifiers since every
2 Sabrina Maniscalco et al.: Transient dynamicsof linear quantumamplifiers
A
linear amplifier or laser undergoes a transient behavior aa†ρ 2a†ρa+ρaa† , (1)
before stabilizing. − 2 −
Our results can be directly applied to describe tran- (cid:2) (cid:3)
with a and a† annihilation and creation operator of the
sients in technologicalapplications basedonlinear ampli-
quantum harmonic oscillator and
fiers.Opticallinearamplifiersareessentialcomponentsof
state-of-the-art optical networks. In order to attain best
2g2
performancesofthenetworks,however,itiscrucialtoan- A = r , (2)
γ2 2
alyze the behavior of linear amplifiers during power tran-
sients causing fast switching on and off of the amplifiers 2g2
C = r , (3)
[8].Althoughthelinearamplifierscurrentlyusedinoptical γ2 1
networksdonotneedtooperateatthequantumlevel,the
recentdevelopmentofquantumtechnologiessuchasquan- wheregisthecouplingstrengthoftheinteractionbetween
tumcommunication,quantumcryptographyandquantum the two-level atoms and the mode of the field, ri = Ni/γ
computation, is basedon the implementation of networks (withi=1,2)is the pumping rateinto the atomicleveli,
containingnanodevicesandnanocomponentsoperatingat and γ is a rate of the same order of the atomic linewidth.
the quantum level. A relevant quantity in the dynamics is the linear gain
Recentlyaschemeforoptimalcloningofcoherentstates (ordamping)factor,describingthe lineargrowth(loss)of
with phase-insensitive linear amplifiers and beams split- energy in the mode,
tershasbeenproposed[9].Recentadvancesinthe fieldof
nanoelectromechanicalsystemshavepavedthewaytothe W =A C. (4)
−
realization of experiments close to achieve the quantum
When W >0, the master equation describes a linear am-
limited detectionandamplification.In[10], positionreso-
plifier, when W < 0 it describes a linear absorber. The
lutionveryclosetothequantumlimitisobtained,demon-
constant A gives the noise provided by the spontaneous
strating the near-ideal performance of a single-electron
emission; this term is present even if the mode energy is
transistorasalinearamplifier.Anotherrecentapplication
initially zero.
of linear amplifiers consists in a method for reconstruct-
The time evolution of the amplitude of the field is de-
ing a multimode entangled state [11]. Also quasiprobabil-
scribed by the equation:
ityfunctionshavebeenshowntobemeasurableviadirect
photodetection of a linear amplified field [12].
a a(t) =G1/2e−iω0t a , (5)
The paper is structured as follows. Section II sum- h iout ≡h i h iin
marizes the results of earlier investigations for easy com-
where a = a(t =0) , ω is the frequency of the radi-
parison with the present results. Section III presents the h iin h i 0
ation field and the gain G is defined as
solution for time dependent coefficients and discusses its
main properties. In Sec. IV we discuss the possibilities to G=eWt. (6)
retain nonclassical features in the output of the amplifier
relating our results to earlier calculations. In Sec. V we Thegainisgreaterthan1forlinearamplifiersandsmaller
discuss the emergence of thermal features in the solution. than 1 for linear attenuators. The solution of the Fokker-
Finally Sec. VI concludes the discussion of the work. Plank equation for the Glauber-Sudarshan P representa-
tionofthedensitymatrix(P function)canbeusedtocal-
culatethetransformationofanyincomingP (α)function
2 Review on phase insensitive narrow band in
by the amplifier:
linear amplifiers
The simplest standard amplifier configuration consists of Pout(α)= dα0P(α,tα0)Pin(α0) (7)
|
an assembly of N two-level atoms, of which N are ex- Z
2
citedandN1areunexcited,interactingwithasingle-mode where
quantumfield.Itisassumedthatthefieldfrequencyisres-
onant with the atomic frequency and that the population 1 α G1/2e−iω0tα 2
0
P(α,tα )= exp − (8)
of the two-level atoms is partly inverted, i.e. N > N . 0
2 1 | πm(t) "−(cid:12) m(t) (cid:12) #
This is the standard model of a laser; its linear operation (cid:12) (cid:12)
regimedescribes anamplifier [1]. Inthe standarddescrip-
istheamplifiertransferfunction[13].Thetimedependent
tion of linear amplifiers it is assumed that N and N are
2 1 width is given by
maintainedapproximatelyconstantintimebysomepump
and loss mechanism.
m(t)=A[G(t) 1]/W. (9)
Starting fromamicroscopicdescriptionofthe interac- −
tionbetweenthe two-levelatomsandthe quantummode, This quantity represents the average photon number of
it is possible to derive the following master equation for the spontaneous emission field [1]. Note that, for a linear
the field mode in the interaction picture [1] amplifier,thegaingrowsasymptoticallytoinfinityfort
→
dρ C ,andsodoesthewidth[3].Foranabsorber,ontheother
= a†aρ 2aρa†+ρa†a ∞hand, the asymptotic value of the width m(t) is finite.
dt − 2 −
(cid:2) (cid:3)
Sabrina Maniscalco et al.: Transient dynamicsof linear quantumamplifiers 3
By using Eqs. (7) and (8) one can calculate the noise case, the time evolution of the field mode is described by
of the output field, defined as the symmetrically ordered a master equation of the same form of Eq. (1), but with
fluctuations of the field mode, [14]: timedependentcoefficientsA(t)andC(t).Similarly,ifone
switches off the external pump, the ratio of the atomic
∆a2 = 1 a†a+aa† a a† populationsN2/N1 willtendtotheBoltzmannfactorand
| |out 2h iout−h iouth iout the amplification process will eventually stop, the system
1 approaching its thermal equilibrium. In the following we
= G∆a2 +m(t) (G 1)
| |in − 2 − consider the first of these two situations, i.e. the onset of
G ∆a2 + , (10) amplification due to the creation of population inversion
≡ | |in A in an initially damping medium. In more detail, we con-
where (cid:0) (cid:1) sider the case in which
1 A+C 1
= 1 , (11) 2g2 eε(t−t0)
A 2 A C − G A(t) = r (t)=A +B, (13)
(cid:18) − (cid:19)(cid:18) (cid:19) γ2 2 eε(t−t0)+e−ε(t−t0)
istheequivalentnoisefactor,oramplifieraddednoise,in-
2g2 e−ε(t−t0)
troduced by Caves [14]. This quantity describes the fluc- C(t) = r (t)=A +B, (14)
tuations of the internal modes of the amplifying medium. γ2 1 eε(t−t0)+e−ε(t−t0)
Since the input field and the internal modes of the am-
where ε is the rate of change of the pumping coefficients,
plifying medium are uncorrelated, their fluctuations add
that is the amplification onset rate. We assume that at
inquadratureandtheyarebothamplified.Theminimum
t= the state of the ensemble of two-levelatoms con-
value of the added noise for infinite gain, e.g. for t , −∞
→∞ stituting the amplifying medium is thermal, that is
is given by the Caves limit:
N (t ) A(t ) B
1 A+C 1 1 2 →−∞ = →−∞ = (15)
AC = 2 A C = 2 +θ ≥ 2, (12) N1(t→−∞) C(t→−∞) A+B
(cid:18) − (cid:19) =e−h¯ω0/kBT,
wheretheexcessnoisefactorθgivestheinitialmeannum-
ber of excitations of the internal modes of the medium, which implies B/A = (eh¯ω0/kBT 1)−1 = nM, with nM
−
and therefore approaches zero when the initial tempera- mean number of excitations of the medium. We assume
ture of the amplifying medium vanishes, T 0. that the state of the amplifying medium practically does
It has been demonstrated that the out→put field of a not change in the time interval <t 0.
−∞ ≤
phaseinsensitivenarrowbandlinearamplifiermaypossess UndertheseconditionstheasymptoticgainfactorW(t)
nonclassicalfeatures onlyif the input fieldis nonclassical. takes the form
However,as the amplifier gain increases,any nonclassical
feature of the light, which was present in the input field, W(t)=A(t) C(t)=Atanh[ε(t t0)]. (16)
− −
tends to be lost. In particular, subPoissonian statistics
Note that, for t < t , W(t) < 0 and the system behaves
and squeezing are lost when the gain G exceeds the value 0
asanabsorber,while for t>t ,W(t)>0 andthe system
2 [1,13]. 0
behavesasanamplifier.Thereforet indicatesthetimeat
In the next section we present a theory describing 0
which the amplification process begins. From Eq. (16) we
the transientregimeofthe amplificationprocess.Inother
infer that the constant A is the asymptotic gain factor.
words, we will drop the assumption that N and N are
2 1 Finally we stress that, for t ,
constant and we will describe the onset of the amplifica- →∞
tion process from an initial damping regime. Our aim is
N (t )/N (t )=A(t )/C(t )
to study the transient dynamics and to investigate how 2 →∞ 1 →∞ →∞ →∞
= (A+B)/B =N (t )/N (t ), (17)
the results for the standard amplifier, described in this 1 2
→−∞ →−∞
section, are modified.
i.e., the population of the exited (ground) state tends
asymptoticallytotheinitialpopulationoftheground(ex-
cited) state.
3 Transient regime of linear amplification
Previous work on linear amplifiers deals with a situa- 3.1 The master equation and its solution
tion in which the amplifying medium, e.g. an assembly
of two-level atoms, satisfies the population inversion con- The master equation describing the transient behavior of
ditionrequiredtoamplifyaninputfield.Inordertoreach theamplificationprocess,intheinteractionpicture,isthe
the invertedpopulation condition it is necessaryto pump following
the atoms from their initial thermal condition till the
point in which N > N . During this transient regime dρ A′(τ)
2 1 = aa†ρ 2a†ρa+ρaa†
the pumping rates to levels 2 and 1 (r2(t) and r1(t)) dτ − 2 −
change with time till they reach a stationary value for C′(τ)(cid:2) (cid:3)
which N /N r /r > 1 (amplification regime). In this a†aρ 2aρa†+ρa†a , (18)
2 1 ∝ 2 1 − 2 −
(cid:2) (cid:3)
4 Sabrina Maniscalco et al.: Transient dynamicsof linear quantumamplifiers
with
0.12
e(τ−τ0)
A′(τ) = A′ +B′, (19) ) 0.1
e(τ−τ0)+e−(τ−τ0)
t(
e−(τ−τ0) G 0.08
C′(τ) = A′ +B′. (20)
e(τ−τ0)+e−(τ−τ0)
0.06
Inthepreviousequationswehaveintroducedtherelevant
0.04
physical dimensionless parameters τ = εt, τ = εt , A′ =
0 0
A/ε and B′ = B/ε. As we will see in the following, the
0.02
parameter A′, which is the ratio between the asymptotic
gainfactorandtherateofonsetoftheamplification,plays
2 4 6 8 t 10
a central role in the system dynamics. Indeed both the
gain G(τ) and the added noise depend crucially on this
parameter. Fig.1. TimeevolutionofG(τ)forA′=2,3,4,5.5(increasing
′
valuesofA correspondtodecreasingthicknessoftheline)and
Following the method developed in [5] we solve the
masterequationgivenbyEq.(18)intermsofthequantum τ0=5.
characteristic function (QCF) [15], defined through the
equation calculations,itturnsoutthattheP functionhasthesame
form of Eq. (8), but with G(τ) given by Eq. (23) and
ρ (τ)= 1 χ (ξ)e(ξ∗a−ξa†)d2ξ. (21) m(τ)=[G(τ) 1]/2+∆(τ).
S 2π τ −
Inserting Eq. (16) into Eq. (23) and carrying out the
Z
integration yields
The solution reads as follows
χτ(ξ)=e−∆(τ)|ξ|2χ0 G1/2(τ)e−i(ω0/ε)τξ , (22) G(τ)= cosh(τ −τ0) A′. (26)
cosh(τ )
(cid:16) (cid:17) (cid:20) 0 (cid:21)
where χ is the QCF of the initial state of the field, ω is
0 0 Itisnotdifficulttoprovethat,forτ =0,andinthelimit
0
the field frequency and the G(τ) is the gain, given by
of infinitely fast onset of the amplification (ε ), the
τW(τ′)dτ′ gain function tends to G(t)=eAt =eWt [see E→q.∞(6)].
G(τ)=e 0 . (23) InFig.1 we plotthe gainG(τ) for four increasingval-
R ues of A′. As clearly shown in the figure for increasing
The quantity ∆(τ), appearing in Eq. (22) is defined as
values of A′ the values of the gain in proximity of the
follows
amplificationtime τ become smaller and smaller. This is
0
1 t because small values of A′ correspond to small values of
∆(τ)= [G(τ)] [G(τ′)]−1[C′(τ′)+A′(τ′)]dτ′. (24) the asymptotic gainfactoror,equivalently,to a very slow
2
Z0 amplificationonsetrate.Ingeneral,thegaindecreasesfor
ItisworthunderliningthatthesolutiongivenbyEq.(22), times smaller than τ0 and, as expected, starts to increase
with the help of Eqs. (23)-(24), holds whatever the ex- after the amplification sets in, even if G(τ) > 1 only for
plicit time dependence of the coefficients A(τ) and C(τ), timesτ >2τ0.Notethatthestandardtheoryoflinearam-
appearing in Eq. (18), is. The case considered in the pa- plification predicts that for a linear amplifier it is always
per[Eqs.(13)-(14)]hasbeenchosentoillustratethetran- G(τ) > 1 (see Sec. 2). However, if one takes into account
sient dynamics in a physically reasonable and well justi- the transient regime characterizing the initial dynamics
fied model. Indeed Eqs. (13)-(14) describe a situation in of every linear amplifier it turns out that there exist an
which from an initial condition in which N > N , the interval of time at which, although W > 0, there is still
1 2
populations of the two-levelsystems constituting the am- no gain. The reasonwhy the gainbecomes greaterthan 1
plifying medium pass smoothly to the inversioncondition only after the time 2τ0 is that for 0 < τ < τ0 the system
N > N necessary for amplification. In passing, we note is in a damping regime, and hence the gain decreases. It
2 1
thatthe hyperbolictangenttime dependenceisoneofthe takes exactly another interval of time τ0 after the onset
most commonly adopted phenomenologicalmodels in the of the amplification process to undo the initial decrease
description of transients of physical systems. in the gain. At τ = 2τ0 we have G(2τ0) = G(0) = 1 af-
Starting from Eq. (22), one can calculate the Wigner ter which the gain increases monotonically. This is a new
function,theGlauber-SudarshanP function,andtheHusimifeature brought to light by our theory.
Q function by means of the relation [15] Let us focus on the quantity ∆(τ). Inserting Eq. (26)
into Eq. (24) we get
Wτ(α,p)= π12 Z−∞∞d2ξχτ(ξ)exp(αξ∗−α∗ξ)e(p|ξ|2/2), ∆(τ) = G(τ)A′+2B′ τ cosh(τ′−τ0) −Ad′τ′
(25) 2 cosh(τ )
Z0(cid:20) 0 (cid:21)
where p = 1,0,1 corresponds to the Q, Wigner, and A′+2B′
P functions,−respectively. In particular, carrying out the G(τ) IA′(τ τ0). (27)
≡ 2 −
Sabrina Maniscalco et al.: Transient dynamicsof linear quantumamplifiers 5
For τ =0 and for each A′ real, with A′ =1, we have
0
6 W
sinhτ 1 12 0.4
IA′(τ) = (A′ 1)(coshτ)A′−1 (28) 0.2 τ
− 10 2 4 6 8
F[1,1 A′/2,3/2 A′/2;(coshτ)2], -0.2
− − al -0.4
withF[1,1 A′/2,3/2 A′/2;(coshτ)2]beingthehyperge- gn 8
ometric fun−ction of th−e variablex=(coshτ)2. For A′ =1 si
6
the integral appearing in Eq. (27) is simply equal to
I1(τ)=2arctan(eτ). (29) 4
The mathematical expression of the added noise in the
2
specialcaseofintegervaluesofA′isdiscussedinAppendix
A. For τ =0, one gets
0 6 τ
A′+2B′ 2 4 6 8
∆(τ)=G(τ) cosh(τ0)A′ Fig. 2. Damping and amplification of an initial input field
2 havingha(t=0)i=10,forτ0 =4,A′=0.5andB′ =0.5·10−2.
[IA′(τ −τ0)+IA′(τ0)], (30) The dashed lines indicate signal width ∆(τ)1/2. The insert
shows the gain factor W(τ) = A(τ)−C(τ) in the same time
where I (τ τ ) and I (τ ) are obtained from Eq. (28)
A′ 0 A′ 0
− intervaland for thesame valuesof theparameters.
by substituting for the variable τ the expressions τ τ
0
−
and τ , respectively.
0
The Caves limit is obtained for an infinitely fast onset of
the amplification process, that is for ε , i.e. A′ 0.
3.2 Noise of the output field
Substituting A′ =0 into Eq. (35), and→rem∞embering→that
From the QCF solution given by Eq. (22) we can easily Γ(1)=1 and Γ(1/2)=√π, one gets
calculate the mean values of observables of interest, e.g.
1 B 1
those characterizing the output field statistics, by means
= + , (36)
C
of the relation [15] A 2 A ≥ 2
a†man = d m d ne|ξ|2/2χ(ξ) . (31) that is the Caves limit [see Eq. (12)]. From Eq. (35) one
h i dξ −dξ∗ can see that, for fixed values of the initial mean number
(cid:18) (cid:19) (cid:18) (cid:19) (cid:12)ξ=0 of excitations of the medium n = B/A, is actually
(cid:12) M C
We look first of all at the symmetrically ord(cid:12)ered fluctua- the smallest asymptotic value of the addedAnoise.
(cid:12)
tion, as defined by Caves [14] A careful analysis of the noise at the output field, as
given by Eq. (32), shows that this quantity, as one would
|∆a|2out = G(τ)|∆a|2in+∆(τ) expect, increases monotonically with time whatever the
= G(τ) ∆a2 + (τ) , (32) initialstateis.InFig.2weshowthedynamicsofaninput
| |in A
field which is damped for τ < τ = 4 and then amplified
0
The added noise is given by(cid:2) (cid:3)
forτ >τ =4.Thefigureshowsthatthewidthofthesig-
0
nalalwaysincreases.Inthefollowingsectionwewillstudy
=∆(τ)/G(τ), (33)
A in more detail the transient dynamics for different types
with ∆(τ) given by Eq. (30). This quantity is clearly dif- ofinputfields.Sincewearedealingwithphaseinsensitive
ferent from the one given by Eq. (11) for the standard amplifiers/absorbers, it turns out that it is not possible
linear amplifier case. It is possible to show that for infi- to generate nonclassical states from classical input fields.
nite gain, e.g. for τ , this quantity is always greater However, it is possible to analyze how the conditions to
→ ∞
orequalto , being the Caveslimitforaninfinitely retaininitial nonclassicalfeaturesare modified due to the
C C
A A
fast onset of the amplification process. transient dynamics. In addition we will look at the field
In order to derive the Caves limit from Eq. (33), we statistics by explicitly calculating the time evolution of
note that for τ , I , as given by Eq. (28), tends to the Wigner function.
A′
→ ∞
[16]
√π Γ(A′/2)
I∞ = . (34)
A′ 2 Γ[(A′+1)/2] 4 Nonclassical properties of the output field
Therefore the asymptotic value of the added noise is
4.1 Squeezing and subPoissonian statistics
A′+2B′ A′+2B′√π Γ(A′/2)
= I
A 2 A′ → 2 2 Γ[(A′+1)/2] Letusbeginstudyingtheconditionsforwhichtheoutput
1 B Γ(A′/2+1) field canretain squeezing properties when the input state
= + √π . (35)
2 A Γ[(A′+1)/2] is a squeezed state of the electromagneticfield. We define
(cid:18) (cid:19)
6 Sabrina Maniscalco et al.: Transient dynamicsof linear quantumamplifiers
the dimensionless quadratures of the field as follows
1 (a)
u = a+a† , (37)
√2 τ
i (cid:0) (cid:1) 0.2 G( Q) = 1.77
v = − a a† . (38)
√2 −
0
(cid:0) (cid:1)
The squeezed states satisfy the minimum uncertainty re- )
τ
lation ∆u∆v = 1/2, but are characterized by an unequal Q(-0.2
distribution of the quantum fluctuations
-0.4
s 1
∆u= , ∆v = , (39) -0.6
√2 √2s
-0.8
with s=1. Introducing the rotating coordinates
6
-1 τ
u˜ = ucos(ω t) vsin(ω t), (40)
0 0
− 0 2 4 6 Q 8
v˜= vcos(ω t)+usin(ω t), (41)
0 0
(b)
and using Eq. (31) we get τ
0.4
(∆u˜)2 =G(τ) (∆u˜)2 + , (42) G( Q) = 1.32
out in A 0.2
(∆v˜)2out =G(τ)(cid:2)(∆v˜)2in+A(cid:3) , (43) 0
with givenbyEqs.(33)and(cid:2)(27).Foran(cid:3)inputsqueezed
A -)0.2
statehavings<1,theoutputcanremainsqueezedifand τ
(
only if Q
-0.4
1
G(τ) (∆u˜)2 + < . (44)
in A 2 -0.6
It is possible to show (cid:2)that the max(cid:3)imum allowed value of
-0.8
the gain G(τ), in order to retain squeezing at the output,
decreases with A′ and B′. In other words, for increasing -1 τ
values of A′ and B′, one can retain squeezing only for 0 0.2 0.4 0.6 Q 0.8 1
smaller and smaller values of the gain (less efficient am- Fig. 3. Mandel parameters of the output fields for an initial
′ ′
plification). Having in mind Eq. (30), one finds that, for Fockstate|n0 =5iinthecases A =0.05 (a),andA =1(b).
τ =0,theoutputfieldisstillsqueezedifthegainsatisfies We have set τ0 =0 and B/A=10−2 in both (a) and (b). We
0
the following inequality indicate with τQ the instant of time at which Q = 0. In the
upper-left corner we indicate the corresponding value of the
1 gain at τ =τ : G(τ ).
G(τ)< . (45) Q Q
s2+(A′+2B′)I (τ)
A′
It is worth recalling the standard result for phase insen- This quantity gives an indication of the statistics of a
sitive linear amplifiers, which states that the upper limit quantized field. For a Fock state Q takes its lowest value
for the gain compatible with squeezing at of the output Q= 1while foracoherentstate Qis equalto0.There-
−
field is G= 2, the magic number for photon cloning [13]. fore, values of Q < 0 indicate subPoissonian statistics,
The analysis of the behavior of the gain in our case is while Q = 0 indicates Poissonian statistics and Q > 0
more complicated, since the r.h.s. of the inequality (45) superPoissonian statistics. Using Eq. (31) we derive the
dependsontime.Anumericalstudyshowsthat,although time evolution of the Mandel parameter as follows
for certain time intervals, the r.h.s of the inequality may
be greater than 2, in these time intervals G(τ) is always Q =hni2out+[G(τ)]2hniin[Qin−hniin], (47)
out
smaller than 2. Hence, also in the case studied in this n
out
h i
paper G=2 constitutes anupper limit for retaining non-
where n and n are the mean number of photons
classical features in the output field. h iout h iin
of the output and input fields, respectively, and
Let us now look at the dynamics of an input Fock
state. We recall that one of the nonclassical features of 1
n =G(τ) n + [G(τ) 1]+∆(τ)
such states is that their statistics is subPoissonian. Simi- out in
h i h i 2 −
larly to what we havedone for the squeezedstates we an-
=G(τ) n +m(τ). (48)
in
alyze the requirements to retain subPoissonian statistics h i
at the output field. To this aim we introduce the Mandel For a Fock input state Q = 1 and n = n , hence
in in 0
− h i
parameter Q [1] the condition for having subPoissonian statistics at the
output is
n2 n 2
Q= h i−n h i −1. (46) n out( n out+1)<G2(τ)n0(n0+1). (49)
h i h i h i
Sabrina Maniscalco et al.: Transient dynamicsof linear quantumamplifiers 7
Anumericalanalysisshowsthat,asforthesqueezing,also ( )
for the subPoissonian statistics, we obtain the limit 2 for
τ=0 a τ=6
thegaintypicalofthestandardtheoryoflinearamplifiers.
As an example, in Fig. 3 we compare the Mandel pa-
7.5
rametersoftheoutputfieldsforaninitialFockstate n =
5 in the cases A′ = 0.05 (fast onset of the amplific|at0ion 5
aind/or small value of the asymptotic gain) and A′ = 1. 2.5
0
From the figure one sees that, for A′ = 0.05, one can re-
-2.5
tainsubPoissonianstatistics of the output field for higher
-5
values of the gain compared to the A′ =1 case.
y-7.5
α τ=12 τ=18
7.5
4.2 Wigner function 5
2.5
Let us now have a look at the complete statistic of the 0
-2.5
output field, by means of the Wigner function. Inserting
-5
Eq. (22) into Eq. (25), and putting p=0 we get
-7.5
W (α)= 1 ∞d2ξe−∆(τ)|ξ|2eαξ∗−α∗ξ -7.5-5-2.50 2.5 5 7.5α -7.5-5-2.50 2.5 5 7.5
τ π2 Z−∞ ( ) x
χ (G1/2(τ)e−iω0τ/εξ). (50) b
0
0.8
Inserting the inverse Fourier transform of Eq. (25) into 0.7
Eq.(50) gives 2)0.6
u
∆
1 ∞ ( 0.5
Wτ(α)= π2 d2α0W0(α0) 0.4
∞ Z−∞ 0.3
d2ξe−∆(τ)|ξ|2eb(τ,α,α0)ξ∗−b∗(τ,α,α0)ξ 0.2
Z−∞ 0.1
= π1 ∞d2α0W0(α0)exph−∆|b((ττ∆,)α(,τα)0)|2i 2 4 6 τ 8
Z−∞ Fig.4. ContourplotsoftheWignerfunctionatdifferenttimes
1 ∞ τ for an inputsqueezed state with r=1.Wehaveputτ0 =4,
d2α W (αα )W (α ), (51) ′
0 τ 0 0 0 A = 0.1, n = B/A = 0.1 (a). Variance of the quadrature u˜
≡ π | B
Z−∞ as a function of time, for the same values of the parameters
(b).
with
b(τ,α,α )=α α G1/2(τ)eiω0τ/ε. (52)
0 0
−
In the derivation of Eq. (51) we have used the property Wenowconsiderthecaseofaninitiallysqueezedstate.
that the Fourier transform of a Gaussian is a Gaussian. The initial QCF for squeezed coherent state is
The quantity W (αα ) is the propagator which, for τ
τ 0
| →
0, tends to the delta function δ(α α ).
0
− 1
If the state of the input field is a coherent state α0 , χ (ξ)=exp ξC ξ∗e−iφS 2+i(ξ∗α∗+ξα ) .
than the Wigner function of the output state read|s ais 0 −2| r − r| 0 0
(cid:20) (cid:21)
follows (54)
Here C = cosh(r) and S = sinh(r) , α is the displace-
r r 0
exp |α0G1/2(τ)eiω0τ/ε−α|2 ment of the input field and z = re−iφ is the squeezing
1 − ∆(τ)+1/2 argument.
W (α)= (cid:20) (cid:21). (53)
τ
π ∆(τ)+1/2 For an input squeezed vacuum state (α = 0), with
0
squeezing angle φ = 0, the Wigner function at time τ
The Wigner function of the output state is therefore a takes the form
Gaussian.Having in mind the time evolution of G(τ) [see
Eq.(26) and Fig.1], one realizes that, in a frame rotating
with the frequency ω0, the Wigner function of an input Wτ(α)= π12 ∞ d2ξe−∆(τ)|ξ|2e(αξ∗−α∗ξ)
coherentstate α0 ,withα0 =0,movestowardsthecenter Z−∞
| i 6
of the phase space for τ < τ and then moves away for 1
0 exp G(τ)e−iω0τ/εξC eiω0τ/εξ∗S 2 . (55)
τ >τ0, while its width continuously increases. −2 | r− r|
(cid:20) (cid:21)
8 Sabrina Maniscalco et al.: Transient dynamicsof linear quantumamplifiers
ThisFouriertransformationcanbecalculatedwiththe
method used in [17], the result being 2
1.01
Wτ(α) = Mexp 2∆(τ)+G(−τ)2(αC2x +S )−1 K)1.8 1N / N21.005 2 4 6 8 τ10 12
(cid:20) 2r 2r (cid:21) e (1.6 0.995
+ exp"2∆(τ)G(τ−)(2Cα22yr −S2r)−1# (56) eratur1.4 0.99
p
m1.2
Here,αx andαy aretherealandimaginarypartsofα,and e
T
M is a time dependent normalization constant. We note 1
thatthisresultisconsistentwiththeEqs.(42)-(43)usedin
Sec. 4.1 to study the time evolution of the quadraturesof 0.8
the fieldfor aninitial input squeezedstate.Contourplots τ
showing the time evolution of the Wigner function are 0 2 4 6 8 10 12
Fig. 5. Time evolution of the temperature of the system for
shown in Fig. 4 (a), while in Fig. 4 (b) we show the time
′ ′
A =1(dottedline)andA =0.05(solidline).Forbothgraph-
evolutionof the squeezing ofthe quadratureamplitudeu˜.
ics we have set ω0 = 1014Hz and nB = B/A = 103. The box
in the top left corner is the ratio between the populations of
′
thetwo-levelatoms. ThisquantitydoesnotdependonA and
5 Departure from thermal equilibrium ′
B separately, but on the ratio n =B/A only [see Eqs. (13)
B
and (14)].
The analytic approach we have described in the previous
section to analyze the onset of the amplification process,
canbeusedtostudyhowasystemdepartsfromaninitial Eq.(58) shows that the medium plus the pumping lasers
thermalequilibriumsituation.Inmoredetail,weconsider behave,asfarasthesystem(modefield)isconcerned,asa
the case in which the medium, modelled as an ensem- thermal reservoir at varying temperature T(τ), as known
ble of two-level atoms, is initially in thermal equilibrium fromthe theory oflaser cooling [20]. Havingin mind that
with the mode of the quantized field. The ratio N /N 2 n +1=coth[h¯ω /k T(τ)][1],andusingtherelation
2 1 out 0 B
betweenthepopulationsoftheexcitedandgroundstates, arhcoith(x)=[ln(x+1) ln(x 1)]/2 (for x2 >1), we can
− −
respectively, is therefore given by the Boltzmann factor express the time evolution of the temperature as follows:
N2/N1 = e−h¯ω0/kBT. The state of the field is a thermal
state at T temperature. T(τ)= ¯hω0 ln[ n +1] ln[ n ] −1. (59)
out out
Atτ =0oneswitchesonpumpinglaserswhichchange k { h i − h i }
B
the population of the two-level atoms until the condition
ofpopulationinversion,necessaryfortheonsetoftheam- Figure 5 compares the behavior of T(τ) for the two cases
plification process, is reached. The pumping lasers alter A′ = 1 and A′ = 0.05. The figure shows clearly that in
the initial condition of equilibrium between the medium bothcasesthe temperature ofthe systemisconstantdur-
andthesystem(thefieldmode).Inordertostudyhowthe ing the damping regime and it starts to increase when
systemdeparts from the conditionof thermalequilibrium approachingthe population inversion at τ =τ0, i.e. when
with the two-level atoms medium, we use the solution of the change in the population ratio N2/N1 becomes con-
the Master Equation (18) to calculate the time evolution siderable.The increasein the temperature is muchhigher
of the Wigner function of the field. For an initial thermal for higher values of A′, since in this case the asymptotic
state, the QCF at time τ has the form gain is also higher.
In order to characterize further the departure from
χ (ξ)=e−∆(τ)|ξ|2exp n + 1 G(τ)ξ 2 . (57) the initial condition of the system we look at the von-
τ out Neumann entropyof the field. We use the result obtained
− h i 2 | |
(cid:20) (cid:18) (cid:19) (cid:21) by Agarwal [19] to calculate the dynamical entropy for a
InsertingthisequationintoEq.(25)onegetsthefollowing state of the form given by Eq.(58)
expression for the Wigner function at time τ
S(τ) = k [ n +1]ln[ n +1]
B out out
{h i h i
1 1 α2 n ln[ n ] . (60)
Wτ(α)= exp | | . (58) − h iout h iout }
π n +1/2 − n +1/2
h iout (cid:20) h iout (cid:21) From direct inspection in the previous equation one sees
Inthe lasttwoequations, n is the numberofpho- that, similarly to the dynamics of the temperature, the
out
h i
tonsoftheoutputfield,asgivenbyEq.(48).The Wigner entropy remains approximately constant for τ < τ and
0
function of Eq. (58) is the Wigner function of a thermal beginstoincreasewhenτ τ .InFig.6weshowhowthe
0
state at a temperature T(τ) which varies with time. The entropy increase rate (in u≃nits of k ) changes with A′; in
B
medium is not in thermal equilibrium anymore, since the the boxinthe topleftcornerthe dynamicsofthe entropy
pumpinglaserschangethetwo-levelatomspopulationun- for three example values of A′ is shown. For increasing
tilthepopulationinversionconditionisreached.However, values of A′, the linear increase in the entropy due to the
Sabrina Maniscalco et al.: Transient dynamicsof linear quantumamplifiers 9
laser pumps up the atoms of the medium till the condi-
2.5 tionofpopulationinversionisreached.Thisisanexample
11S/kB802 AA’=’1=0.5 otifondytnhaamtcicandebpeasrttuudreiedfroamnalayttihcaerllmy.aWl eeqaunilaiblyrziuemthecotnimdie-
kB2 6 A’=0.05 evolutionofthetemperatureandofthevon-Neumannen-
4
τ
∆ 2 tropy on the characteristic parameters of the linear am-
1S/.5 2 6 10 τ 14 plifier. We find that, as known from the theory of laser
∆
cooling, the medium plus the pumping lasers behave, as
far as the system is concerned, as a thermal reservoir at
1
varying temperature. We find that the entropy increase
rate depends crucially on the asymptotic gain.
0.5
7 Acknowledgements
0.5 1 1.5 2
A’
Fig. 6. Dependence of the entropy increase rate on
′ This work has been supported by the European Union’s
A. The entropy increase rate is defined as ∆S/∆τk =
B TransferofKnowledgeprojectCAMEL(GrantNo.MTKD-
[S(τ =14)−S(τ =10)]/4k .Intheboxinthetopleftcorner
B
CT-2004-014427). J.P. and S.M. thank Nikolay Vitanov
thetime evolution of theentropy for the threeexemplary val-
′ ′ ′ for the hospitality at the University of Sofia and Stig
ues A =1, A =0.5, and A =0.05 is shown. In all the plots
Stenholm for the hospitality during the visit to KTH in
we haveset τ0 =8, and nB =B/A=10.
Stockholm. J.P. acknowledges financial support from the
AcademyofFinland(project204777),andfromthe Mag-
amplification process is faster and faster. This result is in nus Ehrnrooth Foundation. S.M. acknowledges financial
accordance with the behavior of the temperature of the support from the Angelo Della Riccia Italian National
system.Infact,smallervaluesofA′ correspondtosmaller Foundation.
asymptotic gain and therefore less efficient amplification
processes.
Appendix A
6 Conclusions
ForintegervaluesofA′ theintegralIA′ definedinEq.(27)
In this paper we have discussed the dynamics of a quan- is given by [18]
tum linear amplifier during the onset of the amplification
process.Foranamplifyingmediumconsistingofanassem- sinhτ 1
I (τ)=
bly oftwo-levelatoms,our theory describes the dynamics 2m 2m 1(coshτ)2m−1
of the output field when the medium passes from a con- −
m−1
dition in which the population of the atoms is thermal, 1+ Γ(m)Γ(m−k−1/2)(coshτ)2k , (61)
to a condition of population inversion characterizing the " Γ(m k)Γ(m 1/2) #
k=1 − −
amplifying regime. X
sinhτ 1
Wehavesolvedexactlythemasterequationdescribing I (τ)=
2m+1 2m (coshτ)2m
the transient dynamics of the linear amplifier in terms of
the quantum characteristic function. The solution is used m−1
Γ(m k)Γ(m+1/2)
to investigate conditions under which an input nonclassi- 1+ − (coshτ)2k
Γ(m)Γ(m k+1/2)
cal field may retain nonclassical features at the output of " k=1 − #
X
thelinearamplifier.Wederivetheanalyticexpressionsfor (2m 1)!!
+ − arctan(sinhτ). (62)
the output noise, as wellas for the squeezing,the Mandel
(2m)!!
parameter and the Wigner function of the output field,
and we use them to characterize completely the transient In this appendix we show that the two equations written
dynamics of the output field. above are special cases of Eq. (28).
Ourresultsarecomparedwithearliertheoriesofphase For A′ = 2m 1 an even integer, the hypergeometric
insensitive linear amplifiers which rely on the assumption function reduces≥to a polynomial of order 1 A′/2
that the population inversion is instantaneously reached, −
i.e. neglecting the transient regime. We show that also m ( m) (b) zk
for a slow onset of amplification, the gain G(τ) has to be F[ m,b,c;z]= − k k , (63)
smaller than 2 (the cloning magic number) in order for − (c)k k!
k=0
X
the output field to retain initial nonclassical properties.
Weconcludethepaperanalyzingthesituationinwhich where
theinitialmodeofthefieldandthetwo-levelatomsmedium Γ(z+k)
(z) = ; (z) =1. (64)
areinthermalequilibriumatT temperature.Anexternal k Γ(z) 0
10 Sabrina Maniscalco et al.: Transient dynamicsof linear quantumamplifiers
Using Eq. (63) and the following properties 14. C.M. Caves, Phys.Rev.D 23, 1817 (1982)
15. S.M. Barnett and P.M. Radmore, Methods in Theoretical
Γ(z+1)=zΓ(z) Quantum Optics (Clarendon Press, Oxford 1997)
πcsc(πz) 16. M. Abramowitz and I. Stegun,Handbook of Mathematical
Γ( z)= , Functions (Dover, NewYork 1965)
− zΓ(z)
− 17. K. Matsuo, Phys.Rev.A 47, 3337 (1993)
18. I.S.GradshteinandI.M.Ryzhik,TablesofIntegrals,Series
equation (28) reduces to Eq.(61).
and Products (Academic Press Inc., San Diego 1994)
For A′ = (2m+1), we obtain Eq. (62) from Eq. (28)
19. G.S. Agarwal, Phys.Rev A 3, 828 (1971)
by using the properties
20. S. Stenholm,J. Opt.Soc. Am. B 2, 1743 (1985)
isinh(τ)F[1,1 A′/2,3/2 A′/2;(coshτ)2]
− − −
= F[1/2 A′/2,1/2,3/2 A′/2;(coshτ)2]
− −
= F[ m,1/2,1 m;(coshτ)2],
− −
and
F[ m,1/2,1 m;(coshτ)2] (65)
− −
1
= Γ(1 m)( coshτ)mPm(isinhτ),
2m − − m
where
dm
Pm(z)=(z2 1)m/2 P (z), (66)
m − dzm m
with P (z) Legendre polinomials.
m
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