Quantum state of an injected TROPO above threshold: purity, Glauber function and photon number distribution

Abstract

In this paper, we investigate several properties of the
full signal-idler-pump mode quantum state generated by a triply
resonant non-degenerate Optical Parametric Oscillator operating
above threshold, with an injected wave on the signal and idler
modes in order to lock the phase diffusion process. We determine
and discuss the spectral purity of this state, which turns out not
to be always equal to 1 even though the three interacting modes
have been taken into account at the quantum level. We have seen
that the purity is essentially dependent on the weak
intensity of the injected light and on an asymmetry of the
synchronization. We then derive the expression of its total
three-mode Glauber P-function, and calculate the joint
signal-idler photon number probability distribution and
investigate their dependence on the injection.

Full text

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arXiv:0707.1565v1 [quant-ph] 11 Jul 2007
EPJ manuscript No.
(will be inserted by the editor)
Quantum state of an injected TROPO above threshold : purity,
Glauber function and photon number distribution
T. Golubeva1, Yu. Golubev1, C. Fabre2, N. Treps2
1V. A. Fock Physics Institute, St. Petersburg State University, 198504 Stary Petershof, St. Petersburg, Russia
2Laboratoire Kastler Brossel, Universit´ e Pierre et Marie-Curie-Paris 6, ENS, CNRS ; 4 place Jussieu, 75005 Paris, France
February 1, 2008
Abstract. In this paper we investigate several properties of the full signal-idler-pump mode quantum state
generated by a triply resonant non-degenerate Optical Parametric Oscillator operating above threshold,
with an injected wave on the signal and idler modes in order to lock the phase diffusion process. We
determine and discuss the spectral purity of this state, which turns out not to be always equal to 1 even
though the three interacting modes have been taken into account at the quantum level. We have seen
that the purity is essentially dependent on the weak intensity of the injected light and on an asymmetry
of the synchronization. We then derive the expression of its total three-mode Glauber P-function, and
calculate the joint signal-idler photon number probability distribution and investigate their dependence on
the injection.
PACS. 4 2.50.Dv, 42.50.Lc
1 Introduction
Three-wave nonlinear interaction in a χ2medium is one of
the main model systems of quantum optics. The problem
is simplified when one inserts an optical cavity around the
non-linear medium, because the resonances of the cavity
permit to restrict the analysis to the intracavity resonant
modes only. The system is called in this case a ”TROPO”
(Triply Resonant Optical Parametric Oscillator). In addi-
tion, in the bellow threshold, one has generally the pos-
sibility to consider the pump as a coherent one with a
fixed classical amplitude and thus to restrict the analy-
sis to a two-mode problem (idler and signal). It has been
shown that such a system produces a squeezed vacuum
state below the oscillation threshold when the two quan-
tum modes are degenerate [1], and twin, quantum inten-
sity correlated beams above the threshold when they are
non-degenerate[2]. Both predictions have been confirmed
over the last two decades by many experiments[3,4]. It has
been also shown that this system has many other interest-
ing quantum properties, especially in the non-degenerate
case: phase anti-correlations[5], and EPR entanglement
below and above threshold[6]. This last feature has been
recently confirmed experimentally[7,8].
In the above-threshold regime, where the three fields
have intensities which are of the same order of magnitude,
the assumption that the intracavity pump field is classical
is not valid. One is faced with a true three-quantum-mode
problem. In such a system, not only a conversion of the
pump wave into the signal and idler waves, but a mutual
conversions of all three modes take place. It was theoreti-
cally shown[5] that under this conditions the pump wave
turns out to be non-classical too. It turns out to be signifi-
cantly squeezed in some parts of the parameter space, and
this was confirmed experimentally[9]. More recently it has
been shown that the pump beam is quantum-correlated
with the sum of the signal and idler fields, and that there
is actually a strong three-partite EPR entanglement be-
tween the three interacting fields[10]. This shows that the
TROPO, which has been one of the most studied systems
by quantum opticians for decades, can still provide us with
good surprises.
Moreover, in the recent years, the attention of the
quantum optics community has gradually shifted from the
determination of squeezed variances and quantum corre-
lations of various observables of the system to more subtle
characterizations and manipulations of its quantum state.
There is for instance a strong development of studies con-
cerning states produced by conditional measurements per-
formed on continuous variable optical systems, such as
photon deletion techniques or selection of instantaneous
values of the photocurrent fluctuations. For such studies,
one needs to know the full quantum state of the system
under consideration, and not only the second order mo-
ment of the observed quantities.
Whereas it is simple to define a quantum state of a
confined optical system, such as a cavity mode or a single
light pulse, it turns out to be very difficult to rigorously
define the quantum state of a c.w. optical beam propagat-
ing from the generating optical device to the detectors,
even though such a concept is often used in the commu-

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2T. Golubeva, Yu. Golubev, C. Fabre, N. Treps: Quantum state of an injected TROPO above threshold
nity from an intuitive point of view. One difficulty that
one encounters is that the state of the system crucially
depends on the exact measurement that one wants to per-
form on it. For example, quantum properties are present
in the system when measured using long integration times,
and the system is perfectly classical for short integration
times.
The purpose of this paper is to bring some insight to
this important issue by considering as an example the ”toy-
model”of the light generated by a TROPO, and to present
various ways of characterizing its full quantum state. We
will consider here only the above threshold case, where all
the generated fields have significant mean values, so that
the linearization method for treating the fluctuations[12,
11] is undoubtedly valid except very close to the oscil-
lation threshold. As is well-known, as far as the signal-
idler phase difference is concerned, a phase diffusion phe-
nomenon takes places, giving rise to diverging phase dif-
ference fluctuations at very long times[13]. To eliminate
this complication, we will assume here that the TROPO
is injected on the signal and/or idler modes, which has the
effect of synchronizing the two fields and locking the phase
diffusion effect. In this case, the validity of the method is
unquestionable, as well as the Gaussian character of all
the output fields.
As a starting point we will determine the full 6×6 co-
variance matrix for the spectral components of the quan-
tum fluctuations of the three output modes. This enables
us to calculate the spectral purity of the TROPO quan-
tum state, and to discuss in which respect the system
can be described or not by a three-field state vector. We
then derive the expression of the intracavity stationary
Glauber function for the whole system. Its interest lies in
the fact that the quasi-probability function contains in a
condensed form all the quantum properties. The way it is
written, and the symmetries that it reveals, are a guide
to determine which are the combinations of the different
modes which have the best quantum properties. We use
it to determine the P-function and purity of the outfield
state in a simple particular case. We finally derive the
joint signal-idler probability distribution, a useful quan-
tity to predict, using a quantum state-reduction approach,
the result of conditional measurements performed on the
system [14].
The article is organized as follows. In section II, we
give the framework of the model that we use to describe
the injected TROPO above threshold. We then derive the
expression of the Fourier components of the quantum fluc-
tuations of the three interacting modes. In the next sec-
tion, we determine and discuss the purity of the system
from the expression of the covariance matrix. We then de-
rive the Glauber stationary P function for the intracavity
fields. We use it to derive the expression of the Glauber
stationary P function for the output fields in the simple
case where the exposure time of the detection is small. In
the last section we derive and discuss the expression of
the joint signal-idler probability distribution as a function
of the detector exposure time. Details of the derivation in
the asymmetrical injection case are given in the appendix
of the paper.
2 Physical model and general equations
2.1 The master equation
A non-degenerate optical parametric oscillator consists of
a non-linear χ(2)medium inserted in a high-Q cavity. This
medium ensures the down-conversionprocess ωp→ ωi+ωs
with exchange of a pump photon with twin signal and
idler photons. As is well known this interaction can be
described in the exact phase matching case by the effective
interaction Hamiltonian [13,15]:
ˆV = i?g
?
= 1,
ˆ apˆ a†
iˆ a†
s− h.c.
l = p,i,s
?
(1)
?
ˆ al,ˆ a†
l
?
In this approach, the three interacting intra-cavity modes
are quantized. The master equation in the interaction pic-
ture for three-mode field density matrix ˆ ρ is :
˙ˆ ρ = −i
?
?ˆV , ˆ ρ
?
−ˆRˆ ρ +
?
m=p,i,s
ˆDmˆ ρ.(2)
The operatorˆR describes the damping of the intracavity
quantum oscillators and its action on the density matrix is
determined by the following equality in the case of exact
resonance between the pump, signal, idler fields and three
cavity modes:
ˆRρ =
?
m=p,i,s
κm
2
?ˆ a†
mˆ amˆ ρ + ˆ ρˆ a†
mˆ am− 2ˆ amρˆ a†
m
?. (3)
where κmis the energy damping rate of the m-mode. The
operatorsˆDm ensure an excitation of each of the actual
modes by a quasi-classical or coherent state of amplitude
?Nin
ˆDmˆ ρ =κm
2
m:
?
Nin
m
?ˆ a†
m− ˆ am, ˆ ρ?
(4)
The external quasi-classical fields with a real amplitudes
?Nin
external coherent waves is used to depress the phase dif-
fusion in the system. For this aim, it would be enough to
put
?Nin
ical and physical situations. Much simpler solutions take
place as
?Nin
on the symmetrical case and giving all wished formulas
for the asymmetrical excitation in App. A.
We can derive from equation (2) the corresponding
evolution equation in the Glauber diagonal representation
mare in resonance with the m-waves.
The excitation of the idler and signal waves by the
i
≡
√Nin?= 0 and
?Nin
s
= 0. However the
asymmetry arising here complicates both the mathemat-
√Nin. In this article we shall
consider both models focusing in the body of the article
i
=
?Nin
s ≡

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T. Golubeva, Yu. Golubev, C. Fabre, N. Treps: Quantum state of an injected TROPO above threshold3
for the three interacting modes. The Glauber function P
is introduced by the integral relation:
ˆ ρ(t) =
? ? ?
P(αp,αi,αs,t) |αp,αi,αs? ×
?αp,αi,αs|d2αpd2αid2αs.(5)
With symmetrical phase locking, the master equation reads:
∂P(αp,αi,αs,t)
∂t
=
?
p)P + g(αiαs∂P
m=i,s
κm
2
∂
∂αm(αm−
√
Nin)P +
+κp
2
∂
∂αp(αp−
∂P
∂αs) + gαp
?
Nin
∂αp
− αpα∗
s
∂P
∂αi
−
−αpα∗
i
∂2P
∂αi∂αs
+ c.c.(6)
The Glauber representation is often un-practical, as, when
non-classical effects are present, the Glauber functions
can be expressed only in terms of distributions, difficult
to handle mathematically. Sudarshan has suggested some
way for writing these distributions [16,17] in the form of
the momentum series, however the master equation re-
mains often impossible to express simply. For instance,
for the simplest model of the sub-Poissonian laser [18],
the master equation contains derivatives of all orders with
respect to complex amplitudes, which means that all mo-
menta are connected with each other in one system of
equations of infinite order. In our problem, the situation
happens to be very favorable as the master equation has
a simple and well defined expression.
2.2 Classical equations for the OPO operation
As is well-known, to get the semi-classical evolution equa-
tions for the mean amplitudes, we can use Eq (6), neglect-
ing the second derivatives with respect to the complex
amplitude. Then one can get:
˙ αp= −κp
˙ αi= −κi
˙ αs= −κs
2
(αp−
(αi−
(αs−
?
√
√
Nin
p) − gαiαs
Nin) + gαpα∗
(7)
2
s
(8)
2
Nin) + gαpα∗
i
(9)
In the following we will only consider the case of equal
losses for the signal and idler cavity losses:
κi= κs≡ κ.(10)
Then it is not difficult to obtain the stationary solutions,
which can be written in the form:
αp=
?Np,αi= αs=
√
N, (11)
where for the values N and Np the following equalities
take place:
g?Np=κ
κ(1 − µ)N = κp(µp− 1)Np.
2(1 − µ),
gN
?Np
=κp
2(µp− 1),
(12)
These equations depend on two dimensionless parameters
µpand µ. µp, the pump parameter, is defined as
µp=
?
Nin
p
Nth
(13)
where Nth= κ2(1 − µ)/(4g2). µp> 1 corresponds to the
above threshold regime. µ, the injection parameter, repre-
sents which fraction of the total signal and/or idler fields
is injected :
µ =
?
Nin
N
. (14)
We will restrict our analysis to µ ≪ 1. There are two
reasons for this. First, we are going to apply the limit
of small photon number fluctuation that is impossible in
the bistability area. Second, for a strong injected field, the
Poissonianstatistics of the external field would be imposed
on the intracavity mode and its quantum properties would
be destroyed.
2.3 Limit of small amplitude and phase fluctuations
We make now two important assumptions. First of all,
because we consider the above threshold situation inside
a high-Q cavity, the limit of the small photon number
fluctuations can be used in our treatment. If we present
the complex amplitudes via amplitudes and phases
αm=√umeiϕm,m = p,i,s,(15)
then we can require
um= Nm+ εm,εm≪ Nm.(16)
Secondly, because of injection, one can find well defined
values for the steady state phases, namely ϕm= 0. We will
assume that in the system the phase fluctuations around
these steady state values are small :
ϕm≪ 1.(17)
Taking into account this limit of small amplitude and
phase fluctuations, one finds that it is possible to factorize
the Glauber distribution in the form:
P(αp,αi,αs,t) =(18)
= P(εp,ε+,t)P(ε−,t)P(ϕp,ϕ+,t)P(ϕ−,t)

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4T. Golubeva, Yu. Golubev, C. Fabre, N. Treps: Quantum state of an injected TROPO above threshold
with decoupled equations for the different factors:
∂P(εp,ε+,t)
∂t
=
?1
∂
∂ε+
+κN(1 − µ)∂2
?
−κN(1 − µ)∂2
?κp
+
∂ϕ+(κ(1 − µ/2)ϕ+− κ(1 − µ)ϕp) −
−κ
∂ϕ2
+
2
∂
∂εp(κpεp+ κ(1 − µ)ε+)+
?µκ
?
∂
∂ε−ε−−
?
∂
∂ϕp(ϕp+ (µp− 1)ϕ+)+
+
2ε+− κp(µp− 1)εp
?
+
∂ε2
+
P(εp,ε+,t),(19)
∂P(ε−,t)
∂t
=κ(1 − µ/2)
∂ε2
−
P(ε−,t), (20)
∂P(ϕp,ϕ+,t)
∂t
=
2
∂
4N(1 − µ)
∂2
?
P(ϕp,ϕ+,t), (21)
∂P(ϕ−,t)
∂t
=
?µκ
2
∂
∂ϕ−ϕ−+
κ
4N(1 − µ)
∂2
∂ϕ2
−
?
P(ϕ−,t).
(22)
Here
ε±= εi± εs,ϕ±= ϕi± ϕs
(23)
One sees that the photon and phase fluctuations turn out
to be statistically independent at exact triple resonance.
Furthermore, the last equation shows that the injected
fields lead to the locking of the differential phase and to
the suppression of the phase diffusion phenomenon present
in the degenerate OPO above threshold. However the syn-
chronizing fields influence the noise properties of the sys-
tem. It is important to understand whether the phase lock-
ing and the presence of significant noise reduction and
quantum correlation are compatible with each other or
not. This is what we will see in the following.
2.4 Intracavity spectral densities
In order to analyze the time dependent correlation func-
tion, let us derive the Langevin equations for the three
interacting fields (19)-(22). They are easily written accord-
ing to well-known rules and have the following form:
˙ εp= −κp/2 εp− κ/2(1 − µ) ε+,
˙ ε+= −κµ/2 ε++ κp(µp− 1) εp+ f+(t),
˙ ε−= −κ(1 − µ/2) ε−+ f−(t),
˙ ϕp= −κp/2 ϕp− κp/2(µp− 1) ϕ+,
˙ ϕ+= −κ(1 − µ/2) ϕ++ κ(1 − µ) ϕp+ g+(t), (28)
˙ ϕ−= −κµ/2 ϕ−+ g−(t),
(24)
(25)
(26)
(27)
(29)
where the stochastic sources are determined by the pair
correlation functions
?f+(t)f+(t′)? = 2κN(1 − µ) δ(t − t′),
?f−(t)f−(t′)? = −2κN(1 − µ) δ(t − t′),
?g+(t)g+(t′)? = −κ(1 − µ)/(2N) δ(t − t′),
?g−(t)g−(t′)? = κ(1 − µ)/(2N) δ(t − t′).
The best way to solve these equations is to rewrite them in
the Fourier domain and solve the simple algebraic system
of equations for the spectral components. One obtains
(30)
(31)
(32)
(33)
(ε2
+)ω= 2N ×
(34)
×
κ(κ2
p+ 4ω2)(1 − µ)
[2ω2− κpκ[µ/2 + (µp− 1)(1 − µ)]]2+ ω2(κp+ κµ)2,
(ε2
p)ω= 2Np×
κ2κp(1 − µ)2(µp− 1)
[2ω2− κpκ[µ/2 + (µp− 1)(1 − µ)]]2+ ω2(κp+ κµ)2,
(εpε+)ω= −2N ×
×
(35)
×
(36)
κ2κp(1 − µ)
[2ω2− κpκ[µ/2 + (µp− 1)(1 − µ)]]2+ ω2(κp+ κµ)2,
(ε2
−)ω= −2N
κ(1 − µ)
κ2(1 − µ/2)2+ ω2,(37)
and
(ϕ2
p)ω= −
1
2Np
×
(38)
×
κ2κp(µp− 1)(1 − µ)2
[2ω2− κκp[µ/2 + µp(1 − µ)]]2+ ω2[κp+ 2κ(1 − µ/2)]2,
(ϕ2
2N×
κ(κ2
[2ω2− κκp[µ/2 + µp(1 − µ)]]2+ ω2[κp+ 2κ(1 − µ/2)]2,
(ϕpϕ+)ω=
2N×
κκ2
p(µp− 1)(1 − µ)
[2ω2− κκp[µ/2 + µp(1 − µ)]]2+ ω2[κp+ 2κ(1 − µ/2)]2,
(ϕ2
−)ω=
2N
+)ω= −1
(39)
×
p+ 4ω2)(1 − µ)
1
(40)
×
1
κ(1 − µ)
(κµ/2)2+ ω2.(41)
In these expressions, the spectral density (ε2
are defined as a factor in front of the delta-functions in the
correlation functions
m)ωand (ϕ2
m)ω
?εm(ω) εm(ω′)? = (ε2
?ϕm(ω) ϕm(ω′)? = (ϕ2
where
m)ωδ(ω + ω′),
m)ωδ(ω + ω′),
m = p,±,
(42)
εm(ω) =
1
√2π
1
√2π
?
εm(t) e−iωtdt,
εm(t) =
?
εm(ω) eiωtdω.(43)

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T. Golubeva, Yu. Golubev, C. Fabre, N. Treps: Quantum state of an injected TROPO above threshold5
The mutual correlations (εpε+)ωand (ϕpϕ+)ωare defined
exactly in the same way:
?ε+(ω) εp(ω′)? = (ε+εp)ωδ(ω + ω′),
?ϕ+(ω) ϕp(ω′)? = (ϕ+ϕp)ωδ(ω + ω′).(44)
3 Spectral purity of the OPO quantum state
3.1 Output field variances and correlations
In the previous section, we have considered the intracavity
spectral densities. Now we want to determine the corre-
sponding quantities for the output beams. We consider
the optimum case where only one mirror of the cavity is
not perfectly reflecting and transmits the light onto the
detectors. Inside the cavity the normalized amplitude was
defined by the photon operators ˆ am(t) (m = p,i,s) that
obey the commutation relations
?ˆ am(t),ˆ a†
m(t)?= δmn.(45)
Let us callˆAm(t) the corresponding operator for the out-
put beams. They obey the commutation relations:
?
?ˆAm(t),ˆAn(t′)
ˆAm(t),ˆAn(t′)†?
= δmnδ(t − t′),
= 0.
?
(46)
and are related to the intracavity ones by the input-output
relations on the coupling mirror:
ˆAm(t) =√κmˆ am(t) −
m = p,i,s.
?
Cm+ˆAm,vac(t)
?
,
(47)
We have taken into account the fact that the reflection co-
efficient of the output mirror is about one. Cpis the com-
plex normalized amplitude of the pump wave in resonance
with the p-mode. This amplitude is related to the quan-
tity Nin
p
introduced earlier by relation Cp=
The amplitudes Csand Ciare the classical injected fields
in resonance with the idler and signal modes: Ci= Cs=
√κNin/2.
ˆAm,vac(t) are the input vacuum fluctuations, with com-
mutation relations
?
κpNin
p/2.
?ˆAm,vac(t),ˆAn,vac(t′)†?
?
We can rewrite (47) for the fluctuations δˆAm=ˆAm−?ˆAm?
and δˆ am= ˆ am− ?ˆ am?, in the Fourier domain:
δˆAm(ω) =√κmδˆ am(ω) −ˆAm,vac(ω).
Let us divide the frequency scale into small equal intervals
of size ∆. Then for each discrete frequency ωl one can
write:
ωl+∆/2
?
ωl−∆/2
= δmnδ(t − t′),
= 0.
ˆAm,vac(t),ˆAn,vac(t′)
?
(48)
(49)
δˆAl
m=
1
√∆
δˆAm(ω) dω.(50)
It is easy to check that the algebra of these operators is
determined by the commutation relations:
?
δˆAl
m,(ˆAk
m)†?
m)†can be thought of as be-
= δlk.(51)
Thus the operators δˆAl
ing the annihilation and creation operators for the photons
in the frequency band around ωl. Then for each mode (fre-
quency) one can introduce the corresponding quadrature
components via the standard relations
m, (ˆAl
δˆ Xl
m=1
2
?
?
(δˆ Al
m)†+ δˆAl
m
?
?
,
δˆYl
m=i
2
(δˆAl
m)†− δˆAl
m
.(52)
Taking into account (49) the variancesof the output quadra-
tures can then be expressed in terms of the intracavity
ones:
?δˆ X2
m?l=1
4+
κm
4Nm
1
∆
ωl+∆/2
?
ωl−∆/2
(ε2
m)ωdω,
?δˆY2
m?l=1
4+ κmNm
1
∆
ωl+∆/2
?
ωl−∆/2
(ϕ2
m)ωdω.(53)
Going to the continuous frequencies by means of ∆ → 0,
one can obtain the wished expression giving the observed
variances and correlation in function of the intracavity
ones:
4(δˆ X2
4(δˆY2
m)ω= 1 + κm/Nm(ε2
m)ω= 1 + 4κmNm(ϕ2
m)ω,
m)ω.(54)
4(δˆ Xmδˆ Xn)ω=
4(δˆYmδˆYn)ω= 4
m,n = p,i,s.
?
?
κm/Nm
?
?
κn/Nn(εmεn)ω,
κmNm
κnNn(ϕmϕn)ω,
(55)
3.2 Spectral purity
Let us now introduce the three-mode spectral covariance
matrix of the OPO. In the present case of exact cavity
resonance for the three modes, the amplitude and phase
fluctuations are independent and this matrix turns out to
be quasi-diagonal:
Mω=
?Mǫ
0
0 Mφ
?
, (56)
where ||Mǫ|| are amplitude and ||Mφ|| are phase 3 × 3
matrices given by:
Mǫ=


4(δX2
4(δXiδXp)ω
4(δXsδXp)ω4(δXsδXi)ω
p)ω
4(δXpδXi)ω4(δXpδXs)ω
4(δX2
i)ω
4(δXiδXs)ω
4(δX2
s)ω

,(57)

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6T. Golubeva, Yu. Golubev, C. Fabre, N. Treps: Quantum state of an injected TROPO above threshold
and analogically ||Mφ|| with help of the variances
4(δYmδYn)ω. The spectral variances are determined as a
factor in front of the delta-function in the correlation re-
lation:
?δXm(ω)δXn(ω′)? = (δXmδXn)ωδ(ω + ω′),
?δYm(ω)δYn(ω′)? = (δYmδYn)ωδ(ω + ω′)
A widely used way of describing the properties of a quan-
tum system at a given noise frequency ω is to assign to it
a quantum state, that we will call ”single noise frequency
state”, described by the density matrix ˆ ρω, that can be
experimentally characterized for example by quantum to-
mography. Let us introduce the spectral purity, given by
(58)
Πω= Trˆ ρ2
ω. (59)
This state is a pure state when Πω = 1, or a statistical
mixture when Πω< 1. In the present case of small fluctua-
tions and input Gaussian states, all quantum fluctuations
have Gaussian statistics. Then the purity is equal to
Πω=
1
√detMω
.(60)
3.3 Spectral purity for a symmetrical injection
In this simple case, we will use the basis of sum and differ-
ence modes m,n = p,± to calculate the determinant and
therefore the purity. The determinant of the covariance
matrix can then be written as the product:
detMω= N−DXDY,(61)
where
N−= 2?δˆ X2
DX= 2?δˆ X2
DY = 2?δˆY2
−?ω2?δˆY2
+?ω4?δˆ X2
+?ω4?δˆY2
−?ω,
p?ω− 4?δˆ X+δˆ Xp?2
p?ω− 4?δˆY+δˆYp?2
ω,
ω,(62)
One can see that N−is related to the uncertainty relation
for the differential variances. DX,Y is the determinant of
the 2 × 2 covariance matrix for the correlated sum and
pump amplitude and phase variances.
In this subsection we are going to consider, as an ex-
ample and for the simplicity of the calculation, the case
κ(µp− 1),κ ≪ κp. According to the previous subsection
the differential variances are derived in the form
κ
2N(ε2
2?δˆ X2
−?ω = 1 +
−)ω=
= 1 −
κ2(1 − µ)
κ2(1 − µ/2)2+ ω2, (63)
2?δˆY2
−?ω= 1 + 2κN(ϕ2
−)ω= 1 +
κ2(1 − µ)
(κµ/2)2+ ω2.(64)
From these expressions, one retrieves first a striking but
well-knownresult, characteristicof the non-degenerateOPO
above threshold: N−= 1 for any values of the parameters
such as the frequency, µp, and µ.
In order to get the other determinants DX,Y we need
to derive the corresponding variances in the explicit form:
2?δˆ X2
+?ω = 1 +
κ
2N(ε2
+)ω=
= 1 +
κ2(1 − µ)
κ2[µ/2 + (1 − µ)(µp− 1)]2+ ω2,(65)
+?ω = 1 + 2κN(ϕ2
= 1 −
p?ω = 1 +κp
2κ2(µp− 1)(1 − µ)2
κ2[µ/2 + (1 − µ)(µp− 1)]2+ ω2,(67)
4?δˆY2
2κ2(1 − µ)2(µp− 1)
κ2[µ/2 + (1 − µ)µp]2+ ω2,
2?δˆY2
+)ω=
κ2(1 − µ)
κ2[µ/2 + (1 − µ)µp]2+ ω2,
Np(ε2
(66)
4?δˆ X2
p)ω=
= 1 +
p?ω = 1 + 4κpNp(ϕ2
p)ω=
= 1 −
(68)
2?δˆ Xpδˆ X+?ω=
?κp
2Np
?
κ2?(1 − µ)(µp− 1)
?2κpNp
κ2?(µp− 1)(1 − µ)3
κ
2N(εpε+)ω=
= −
κ2[µ/2 + (µp− 1)(1 − µ)]2+ ω2, (69)
√
2κN (ϕpϕ+)ω=
2?δˆYpδˆY+?ω=
=
κ2[µ/2 + µp(1 − µ)]2+ ω2. (70)
Let us remark that, in contrast to N−, the quantities
N+= 2?δˆ X2
Np= 4?δˆ X2
are non minimum, especially near zero frequencies when
the pump is not strongly above threshold. For example,
when µp− 1 ≫ µ and ω = 0
N+= Np=(µp− 1)2+ 1
+?ω2?δˆY2
p?ω4?δˆY2
+?ω,
p?ω
(71)
µ2
p
µp+ 1
µp− 1
(72)
N+and Nptake the minimum value compatible with the
Heisenberg inequality when the OPO is strongly above
threshold, but take very large values close to threshold.
Figure 1a gives the spectral purity of the three-mode
OPO for different values of the injection parameter (µ =
0,1) and of the pump parameter (µp= 4,2 and 1,1). One
first observes that the spectral purity is equal to one out-
side the cavity bandwidth in all configurations, and also
well above threshold at all frequencies. In contrast, the
spectral purity at zero frequency turns out to be less than
one near threshold (µp= 1,1).

Page 7

T. Golubeva, Yu. Golubev, C. Fabre, N. Treps: Quantum state of an injected TROPO above threshold7
We are therefore led to the conclusion that, close to
threshold, the three-mode ”single noise frequency state”de-
scribing the OPO is mixed at low noise frequencies. Con-
sidering that we have taken into account in a quantum
way all the interacting modes, and that all the intracav-
ity modes are transmitted onto the detectors without any
losses, so that the total input-output evolution matrix is
unitary, this result is somewhat unexpected. But we must
recall that we discuss here the purity not of the complete
system, but the purity of some of its spectral components.
Obviously, the intracavity parametric interaction does not
change the purity of the system as a whole. However it
leads to a redistribution of fluctuations and correlations
between the spectral components of the pump, signal and
idler modes, and as a result the purity of each spectral
component does not generally survive. Such a redistri-
bution is maximum when the coupling is maximum, i.e.
at zero noise frequencies, and vanishes outside the cavity
bandwidth. Figure 1b shows the spectral purity for larger
injection. One can see that in this case the purity of the
system is determined by the statistics of the injected field
and becomes larger.
The non-degenerate OPO is often considered as a two-
mode quantum system, for which the pump can be treated
as a classical quantity. In this point of view, it is described
by a two-mode quantum state, characterized by a density
matrix which is the partial trace of ˆ ρω over the pump
mode. Its spectral purity is related to the determinant of
the two-mode covariance matrix. Such a ”partial purity”
is displayed in figure 2. One observes that it is always
larger than the purity of the total system. This can seem
surprising that subsystem purity is larger that total purity,
but it depends on the presence or absence of quantum
correlations between parts of the system. For instance,
when one only considers the differential mode, by tracing
over the pump and sum modes, one finds that its partial
purity is 1, and therefore that it is in a pure state. It is
thereby possible to extract from our initial mixed state
a pure subsystem by adequately eliminating the modes
responsible for the ”impurity”of the total state
3.4 Spectral purity for an asymmetrical injection
In the previous subsection we have discussed the case of
the symmetrical injection of the OPO when both the idler
and signal modes with equal spectral widths κi = κs
are excited equally by the external fields with amplitudes
?Nin
tions can be found in the appendix of this paper, where
the physical situation with Nin
considered. In this case, the mean output signal and idler
fields are no longer equal. In order to retrieve some sym-
metry we further require that κs = κi(1 − µ), so that
the equality Ni = Ns ≡ N survives for the stationary
situation. Nevertheless the intracavity variances turn out
to be different and essentially differently dependent on µ
than under the symmetrical phase locking (see App A).
Besides, the relation between the variances of the output
i
=
?Nin
s. Here we want to briefly discuss the role of
the asymmetry in the injection. The corresponding deriva-
i
= Ninand Nin
s
= 0 is
beam and the intracavity variances turns out to be more
complicated for the ±-variances:
2?δˆ X2
+(ε2
∓)ω(1 −
2?δˆY2
+(ϕ2
∓)ω(1 −
Fig. 3a shows the spectral dependence of the spectral
purity in the asymmetrical case for a small injection field
parameter µ = 0.1. One can see first that outside the spec-
tral band κ the purity is close to 1, as in the symmetrical
case. The band where purity Πω?= 1, depends on the dis-
tance to threshold and decreases when the pump power
increases. Well above threshold (µp > 2) it depends on
µ, and decreases when the injecting field decreases. (see
figure 3b)
However it is important to stress here that for the zero
frequency we can never neglect the influence of the injec-
tion field even for very small values µ. In the limit of small
µ and µp> 5 we get Πω=0= 1/2.
±?ω= 1 +
κi
8N[(ε2
±)ω(1 +?1 − µ)2+
?1 − µ)2],
?1 − µ)2].
(73)
±?ω= 1 + κiN/2[(ϕ2
±)ω(1 +?1 − µ)2+
(74)
3.5 Multi-frequency squeezing and entanglement in
presence of small injection
We can use formulas (63)-(70) to investigate the squeezing
and entanglement in the OPO in presence of small injec-
tion. Even if this is not new in substance, we prove here
that the presence of a small injected field do not destroy
the non-classical features of the output fields.
Let us go back to the symmetrical case. Because
4?δˆ X2
i,s?ω= ?δˆ X2
+?ω+ ?δˆ X2
−?ω
(75)
one has
4?δˆ X2
= 1 +1
i,s?ω=
2
κ2
κ2(µp− 1 + µ/4)2+ ω2−1
2
κ2
κ2+ ω2. (76)
Strongly above threshold µp ≫ 1 the second term turns
out to be negligible and the maximum squeezing is reached
in the idler and signal waves.
4?δˆ X2
i,s?ω=0=1
2.(77)
We can expect squeezing for µp> 2.
In contrast with the amplitude squeezing the phase
squeezing is found only in the pump mode:
4?δˆY2
p?ω= 1 −2κ2(µp− 1)
κ2µ2
p+ ω2.(78)
On the zero frequency
4?δˆY2
p?ω=0=(µp− 1)2+ 1
µ2
p
.(79)

Page 8

8T. Golubeva, Yu. Golubev, C. Fabre, N. Treps: Quantum state of an injected TROPO above threshold
It is not difficult to see that maximum squeezing 4?δˆY2
1/2 is reached as µp= 2.
In order to evaluate an entanglement of the idler and
signal waves, we use the Duan criterium
p?ω=0=
2?δˆ X2
−?, 2?δˆY2
+? < 1.(80)
Taking into account (63) and (66) one can find that this
criterium is carried out independently of pump. Even if,
strongly above threshold (µp≫ 1) 2?δˆY2
+ω=0? → 1.
4 Stationary Glauber quasi-probability
distribution
In the previous section we considered the properties of
the ND-OPO from the point of view of its spectral com-
ponents, by solving non stationary equations in Fourier
space. Stationary master equations also contain important
pieces of information about the quantum system. We de-
vote this paragraph to the determination of the Glauber
quasi-probability distribution of the present three-mode
system in the limit of small amplitude and phase fluctu-
ations which is relevant for the case of the injected ND-
OPO. It is obtained by putting all time derivatives to zero
in Eqs (19)-(22). We will discuss in the two following sub-
sections the solutions of the amplitude and phase equa-
tions respectively.
4.1 Amplitude quasi-probability distribution
The stationary amplitude quasi-probability P(εp,εi,εs)
can be factorized in the form
P(εp,εi,εs) = P(εp,ε+)P(ε−),(81)
where each of the factors obeys its own equations:
?
κ(1 − µ/2)
∂
∂ε−ε−− κN(1 − µ)∂2
∂ε2
−
?
P(ε−) = 0,
(82)
?1
− κp(µp− 1)εp) + κN(1 − µ)∂2
2
∂
∂εp(κpεp+ κ(1 − µ)ε+) +
∂
∂ε+(κµ/2 ε+−
?
∂ε2
+
P(εp,ε+) = 0.
(83)
Let us first consider the second equation. The solution
has the Gaussian form :
P(εp,ε+) =
1
2π?DpD+
exp
?
−(εp− aε+)2
2Dp
−
ε2
2D+
+
?
(84)
.
The unknown parameters Dp,D+and a are coupled with
the variances by means of the following relations :
D+= ε2
+,Dp= ε2
p−ε+εp2
ε2
+
,a =ε+εp
ε2
+
. (85)
As the equation (83) provides us with a possibility to find
the variances in the explicit form, then taking into account
the previous inequalities we can calculate the parameters
of the distribution Dp,D+and a. From eq. (83) one can
then obtain the algebraic system of equation for the vari-
ances in the form :
−κpε2
−1
p− κ(1 − µ) εpε+= 0,
2(κp+ κµ) εpε+−1
(86)
2κ(1 − µ) ε2
++ κp(µp− 1) ε2
p= 0,
(87)
−κµ ε2
Solving this system, one can get the variances :
++ 2κp(µp− 1) εpε++ 2κN(1 − µ) = 0, (88)
ε2
+= N
1
µ/2 + µp− 1
?
1 +
2κ
κp+ κµ(µp− 1)
κp
κp+ κµ(µp− 1),
?
, (89)
ε+εp= −Np
1
µ/2 + µp− 1
1
µ/2 + µp− 1
(90)
ε2
p= Np
κ
κp+ κµ(µp− 1).(91)
In the limit κ,κ(µp− 1) ≪ κpand µ ≪ 1 these variances
read
ε2
+= N
1
µ/2 + µp− 1,
µp− 1
µ/2 + µp− 1
µ/2 + µp− 1.
ε2
p= Np
κ
κp, (92)
ε+εp= −Np
µp− 1
The knowledge of the variances provides us with a
possibility to find the parameters of the Gaussian quasi-
probability:
D+= N
1
µ/2 + µp− 1,
µp− 1
µ/2 + µp− 1
κp.
Dp= Np
µκ
κp,(93)
a = −κ
Let us discuss now the quasi-probability P(ε−) that is a
solution of equation (82). This equation has a normalized
solution only in the form of the distribution. This is con-
nected with the minus sign in front of the second derivative
with respect to ε−. On the one hand this minus sign in-
forms us about the quantum effects in the generation and
on the other hand it makes impossible the derivation of
the solution as a well-behaved function. The direct result
of this situation is the variance ε2
−turns out to be negative
ε2
−= −N(94)
although the following relation
ε2k
−= (2k − 1)!! ε2
−
k, (95)

Page 9

T. Golubeva, Yu. Golubev, C. Fabre, N. Treps: Quantum state of an injected TROPO above threshold9
typical for the Gaussian distribution, survives. Knowing
the variances allows us to derive the P-function as the
formal series [16,17]:
P(ε−) =
∞
?
k=0
1
k!
?
−N
2
?kd2k
dε2k
−
δ(ε−).(96)
It is also possible to use another formal equivalent expres-
sion:
P(ε−) = exp
?
−N
2
d2
dε2
−
?
δ(ε−)(97)
From these expressions, one can derive Fano parameters
for each of the intracavity modes:
Fi,s=1
4
2 − µp
µ/2 + µp− 1+ 1,Fp= 1.(98)
4.2 Phase quasi-probability distribution
As for the amplitude, the quasi-probability distribution of
the three phases can be factorized in the form:
P(ϕp,ϕi,ϕs) = P(ϕp,ϕ+)P(ϕ−).(99)
The corresponding master equation read (19)-(22):
?κp
2
∂
∂ϕp(ϕp+ (µp− 1)ϕ+) +
∂
∂ϕ+(κ(1 − µ/2)ϕ+−
∂2
∂ϕ2
+
∂2
∂ϕ2
−
−κ(1 − µ)ϕp) −
?µκ
Again the minus sign in front of the last term means quan-
tum phase features in the field. A formal solution of the
equation can be presented in the form of the distribution:
κ
4N(1 − µ)
?
P(ϕp,ϕ+), (100)
2
∂
∂ϕ−ϕ−+
κ
4N(1 − µ)
?
P(ϕ−). (101)
P(ϕp,ϕ+) =
∞
?
m,n=0
(−1)m+n
m! n!
Mmn
dm
dϕm
p
dn
dϕn
+
δ(ϕp) δ(ϕ+),
Mmn= ϕm
pϕn
+
(102)
It is possible to demonstrate that the momenta Mmnare
different from zero only as m + n is the even number.
Then, with help of equation (100), we are able to derive
a recurrence relation connecting different non-zero even
momenta with each other:
−(m κp/2 + n κ)Mmn− m (µp− 1)κp/2 Mm−1n+1+
+nκMm+1n−1− n(n − 1)κ/(4N) Mmn−2= 0.
(103)
In particular, as κ,(µp− 1)κ ≪ κp
ϕ2
p= −
+= −1
1
4Np
µp− 1
µp
1
µp,
1
µp
κ
κp,
ϕ2
4N
(104)
ϕpϕ+=
1
4Np
κ
κp.
As for Eq (101) it describes a stationary Gaussian process
and its solution is well-known:
P(ϕ−) =
1
?π/(µN)
exp
?
−
ϕ2
−
1/(µN)
?
(105)
4.3 Stationary output quasi-probability distribution
and purity
In the previous section, the output beams of the ND-OPO
were presented as a set of mutually coupled field oscilla-
tors with different frequencies. In contrast, the P-function
that we have just derived describes the system in its sta-
tionary state, but only for the three intracavity oscilla-
tors. Unfortunately, to the best of our knowledge, there is
no simple relation between the intracavity P-function and
some kind of quasi-probability distribution describing the
system of the three OPO-modes when they have escaped
the cavity. However it is possible to maintain a single-
oscillator description of the three modes outside the cavity
in a very specific case, as it has been explained by one of
us Ref [20]. The distribution function then only concerns
output fields contained in ”thin layers”of the propagation
axis. The thickness of each of these layers must be much
bigger than the wavelength, but much less than the cavity
correlation length l0 = cτ, where τ the correlation time
of the intracavity field. Then it is formally possible to in-
troduce photon operators ˆ amand ˆ a†
such as?ˆ am,ˆ a†
oscillators and that the propagation of the light in free
space can be presented as a transfer of an excitation from
one of the oscillators to the nearest one along the beam.
These observables correspond to measurements performed
on the output beams that integrate the photocurrent fluc-
tuations over time scales much smaller than τ. One can
then define a quasi-probability distribution Poutfor such
output oscillators, which is related to the already calcu-
lated intracavity P-function by means of
macting in m-th layer
n
?= δmn. This means that the field outside
the cavity can be presented as a set of spatially located
Pout(αp,out,αi,out,αs,out) =
?
=
1
TpTiTs
P
αp,out
?Tp
,αi,out
√Ti
,αs,out
√Ts
?
. (106)
√Ti=√Ts≡
efficients of the coupling mirror for the signal and idler
modes;
?Tp the transmission coefficient for the pump
amplitudes, which are the eigenvalues of the annihilation
photon operator.
Knowing Poutof the whole system outside the cavity,
at least in the restricted meaning described in the previous
paragraph, it is now possible to calculate the purity of the
stationary state of this system. It is the product of four
factors:
√T being the amplitude transmission co-
mode; αp,out,αi,outαs,out are the corresponding complex
Πst= Π1× Π2× Π3× Π4,(107)

Page 10

10T. Golubeva, Yu. Golubev, C. Fabre, N. Treps: Quantum state of an injected TROPO above threshold
where the amplitude factors are
Π1 =
? ?
dεpdε+Pout(εp,ε+)
?
−(ε+− ε′
dε′
pdε′
+Pout(ε′
p,ε′
+) ×
×exp
?
−(εp− ε′
4NpTp
p)2
?
exp
?
+)2
8NT
?
,(108)
Π2 =
?
dε−Pout(ε−)
?
dε′
−Pout(ε′
−) ×
×exp
?
−(ε−− ε′
−)2
8NT
?
, (109)
and the phase ones are:
Π3 =
??
dϕpdϕ+Pout(ϕp,ϕ+)
??
?
dϕ′
pdϕ′
+Pout(ϕ′
+,ϕ′
p)×
×exp
?
−(ϕp− ϕ′
1/(NpTp)
p)2
?
exp
−(ϕ+− ϕ′
2/(NT)
+)2
?
,(110)
Π4 =
?
dϕ−Pout(ϕ−)
?
dϕ′
−Pout(ϕ′
−) ×
×exp
?
−(ϕ−− ϕ′
2/(NT)
−)2
?
.(111)
Substituting Eqs. (84), (96), (102), and (105) and consid-
ering the limit κ, (µp− 1)κ ≪ κpwe get
Π1=
1
(1 + ν T/µ)1/2,
Π2= Π3= 1,(112)
Π4=
1
(1 + T/µ)1/2,
where
ν =
µ(µp− 1/2)
µ/2 + µp− 1,
(113)
If the output mirror of the cavity is highly reflecting, so
that that T ≪ µ, then
Πst= 1. (114)
A similar calculation performed with the intracavity P-
function with the same parameters gives Π1= Π2= Π3=
1, and Π4=√µ, so that the purity of the intracavity state,
Π = µ1/2is much smaller than 1.
Field fluctuations integrated over short time intervals
depend only on the high frequency fluctuations of the out-
put fields. The property that Πst= 1 is therefore certainly
connected to the already noticed feature that Πω= 1 for
noise frequencies larger than the cavity bandwidth.
5 Stationary photon number probability
Another important way of characterizing the quantum
state produced by the TROPO is to determine the full
photon number probability distribution in each of the three
beams exiting the system. This quantity is important to
know, for example when one wants to determine the quan-
tum state which is produced by a conditional measurement
performed on the intensity of one of the output beams.
This is what we will do in this section. More precisely,
we will determine the probability for n photons to cross
the cross section of the output beam during a given time
τ. Let us stress that this function is essentially different
than the already calculated stationary probability for the
photon number because it includes the integration time τ.
Let us first calculate the photon number probability
inside the cavity. The joint probability Cin(ni,ns) to find
ni photons in the idler intracavity mode and simultane-
ously nsphotons in the signal intracavity mode is defined
as
Cin(ni,ns) =
?
np=0,1,...
?npnins|ˆ ρ|npnsni?,(115)
where ˆ ρ is the stationary three-mode intracavity density
matrix. By using the Glauber diagonal representation (7)
in (78), we have
Cin(ni,ns) =1
4
∞
?
0
∞
?
0
duidusPred.(ui,us) ×
×e−uiuni
i
ni!e−usuns
s
ns!, (116)
where Pred.is a ”reduced”Glauber photon quasi-probability
given by
Pred.(ui,us) =
??
d2αp
??
dϕidϕsP(αp,αi,αs),(117)
where αl =√ul exp(iϕl)(l = s,i). From the previous
section
Pred.(ui,us) = P(ε−)
?
+
dεpP(εp,ε+) = (118)
=
1
?2πD+
exp
?
−
ε2
2D+
?
exp
?
−N
2
d2
dε2
−
?
δ(ε−)
Substituting this into (116) one can get after the corre-
sponding integrating one can obtain that
Cin(ni,ns) = 1/
√
2πλN exp
?
−(n+− 2N)2
2λN
−n2
2N
?
×
×1/
√
2πN exp
?
−
?
, (119)
where
λ = 2 +
1
µ/4 + µp− 1
(120)
(κ, (µp− 1)κ ≪ κp).
This function describes the probabilities to find niphotons
in the idler mode and nsphotons in the signal mode inside

Page 11

T. Golubeva, Yu. Golubev, C. Fabre, N. Treps: Quantum state of an injected TROPO above threshold 11
the cavity in the stationary regime. Although knowing the
intracavity probability is not enough for determining the
probability for the output travelling fields, it will helps us
to guess the general form of the wished function. We will
assume that the outside probability for output fields has
the same Gaussian form as the probability for the intra-
cavity fields, but with its own parameters depending, in
particular, on observation time τ. We will therefore write
it as:
Cout(ni,ns) = 1/?2πd+ exp
×1/?2d−exp
where the parameters Nout, d± depend on τ. The way
to calculate the parameters will be demonstrated on the
simpler photon probability of a single mode.
By integrating (121) over one of the variances εior εs
we obtain the single-mode photon number probability for
both the idler and signal modes:
?
−(n+− 2Nout)2
2d+
−n2
2d−
?
×
?
−
?
.(121)
Cin(nl) =
1
√2πNFin
exp
?
−(nl− N)2
2NFin
?
, (122)
Fin=1
4(1 + λ),l = i,s,
and correspondingly for the output beam:
Cout(nl) =
1
√2πFout
exp
?
−(nl− Nout)2
2NFout
?
. (123)
In the last probability nlare eigen-numbers of the operator
ˆ nl(t) =
t+τ/2
?
t−τ/2
ˆA†
l(t′)ˆAl(t′) dt′, (124)
where the operatorsˆAlandˆA†
in free space (see (46)). Then
ldescribe light propagating
Nout=
t+τ/2
?
t−τ/2
?ˆA†
l(t′)ˆAl(t′)? dt′= κτN.(125)
In order to determine the Fano-factor Fout, we have to
calculate the variance
?ˆ n2
l? − ?ˆ nl?2= NoutFout.(126)
For the stationary flux
NoutFout=(127)
=
τ/2
?
−τ/2
τ/2
?
−τ/2
dt1dt2?ˆA†
l(t1)ˆ Al(t1)ˆA†
l(t2)ˆAl(t2)? − N2
out.
Following to standard procedure, one obtains
Fout= 1 + κ/N
?
dω (ε2
l)ωδτ(ω),(128)
where the integral contains a product of two functions.
One of them is
κ/N (ε2
l)ω=1
2
?
κ2
(µ/2 + µp− 1)2κ2+ ω2−
κ2
κ2+ ω2
?
.
(129)
It is the spectral density of the amplitude noise in the
selected l-wave that can be obtained from (35) and (37).
The other is
δτ(ω) =1
π
sin2ωτ/2
τω2/2
.(130)
For long enough integration times τ, δτ(ω) gets close to a
δ-function when τ → ∞, so that Foutis given by
Fout=1
2
µp
µ/2 + µp− 1.(131)
Near threshold Fout = µ−1≫ 1, so that the photon
statistics of the signal or idler field turns out to be super-
Poissonian. In contrast, strongly above threshold Fout=
1/2, and the photon statistics of the signal or idler field is
now sub-Poissonian, as it has been already noticed (ref ?)
and experimentally checked (Kasai) .
The joint signal-idler photon probability distribution
can be derived exactly in the same way, knowing the two-
mode variances d±(121):
d±= 2Nout
?
1 + κ/(2N)
?
dω (ε2
±)ωδτ(ω)
?
.(132)
As (µp− 1)κ,κ ≪ κp
κ/(2N) (ε2
+)ω=
κ2
(µ/4 + µp− 1)2κ2+ ω2,
κ2
κ2+ ω2
κ/(2N) (ε2
−)ω= −
(133)
Inserting these expressions to formula (132), one obtains
the full joint photon number probability distribution Cout(ni,ns)
of the output signal and idler beams for any observation
time τ.
If we choose the short observation time τ such as
κτ, κτ(µ/4 + µp− 1) ≪ 1 the variances read:
d±= 2Nout. (134)
This means that under the observation for only short time
the photon statistics turn out to be Poissonian.
On the other hand, for κτ, κτ(µ/4 + µp− 1) ≫ 1,
d+= µp/(µ/2 + µp− 1),d−= 0. (135)

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12T. Golubeva, Yu. Golubev, C. Fabre, N. Treps: Quantum state of an injected TROPO above threshold
We have a possibility to derive the wished photon number
probability in form:
Cout(ni,ns) =
1
?2πd+
So if we count photons during long time intervals, the
photon numbers in both waves turn out to be the same.
This the form of the joint probability distribution that was
guessed in [22], and used to calculate the state of the signal
mode produced by a conditional measurement performed
on the idler mode.
(136)
=
exp
?
−(2ni+ −2Nout)2
2d+
?
δ(ni− ns).
6 Conclusion
One of our aims here was to investigate the quantum
state purity of the TROPO radiation. There is a widely-
distributed opinion that the purity of the output emission
state must be equal to one. The reason of that is con-
nected with a representation about the OPO as system
that, in the unitary process, converts the pure state of the
input light (the pump wave is in the coherent state and
the other modes are in the vacuum states or the pump,
idler, signal waves are in the coherent states and again
the other modes are in the vacuum states) into the same
pure state of the output light. This conclusion absolutely
correct for the output field as whole nevertheless does not
concern the states for the selected frequencies, and just it
is an object of the investigation in experiments and theo-
ries. For example, the covariance matrix is constructed as
a combination of the quadrature variances for the selected
frequency. The purity for this oscillator turns out to be
uncertain and, in principle, can accept any meanings.
We have considered the purity and been convinced
that, strictly speaking, the purity for the state with se-
lected frequency is not one (especially near zero frequen-
cies). However for the symmetrical synchronization the
purity turns out to be close to one, provided the pump
power is strongly above threshold. At the same time un-
der the asymmetrical synchronization the purity becomes
1/2 even if µp ≫ 1. So we can conclude that, generally
speaking, the purity for the oscillator on the selected fre-
quency is not one and essentially depends on the power
of the synchronizing field and the asymmetry of the syn-
chronization too.
We needed to introduce the synchronization into our
the OPO system to depress the diffusion of the differen-
tial phase between signal and idler waves. Usually, it is
suggested that the synchronization especially by the weak
external field does not insert any essential distortions into
a statistical pattern of the TROPO except a phase diffu-
sion depression. However, strictly speaking, this is quite
not obvious and one of our aims here was to investigate
just this side of the OPO generation. As was mentioned
already the purity can be essentially dependent on the
synchronization.
Besides although we introduce the synchronization to
depress the phase diffusion, neverthelessnot only the phase
fluctuations are stabilized under an influence of the exter-
nal field but the amplitude ones too. We have found that
this phenomenon turns out to be essential near threshold
as µp− 1 < µ. One can see that all amplitude variances
on the zero frequency are proportional to 1/µ2(not to in-
finity as under the phase diffusion). This apiaries as well
as in the spectral and stationary variances.
7 Acknowledgements
This work was performed within the French-Russian co-
operation program ”Lasers and Advanced Optical Infor-
mation Technologies”with financial support from the fol-
lowing organizations: INTAS (Grant No. 7904),YS-INTAS
(Grant No. 6078), RFBR (Grant No. 05-02-19646), and
Ministry of education and science of RF (Grant No. RNP
2.1.1.362).
References
1. G. Milburn and D.F. Walls, Opt. Comm. 39, 401 (1981)
2. S. Reynaud, C. Fabre and E. Giacobino, JOSA B 4, 1520
(1987)
3. L. Wu, H. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett.
57 2520 (1986).
4. A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino,
C. Fabre and G. Camy, Phys. Rev. Lett. 59, 2555 (1987)
5. C Fabre, E Giacobino, A Heidmann, L Lugiato, S Rey-
naud, M Vadacchino and Wang Kaige, Quantum Opt. 2,
159 (1990)
6. M.D. Reid and P.D. Drummond, Phys. Rev. Lett. 60,
2731 (1988)
7. Z.Y. Ou, S.F. Pereira, H.J. Kimble and K.C. Peng, Phys.
Rev. Lett. 68, 3663 (1992)
8. A. S. Villar, L. S. Cruz, K. N. Cassemiro, M. Martinelli,
and P. Nussenzveig, Phys. Rev. Lett. 95, 243603 (2005)
9. K. Kasai, G. Jiang and C. Fabre, Europhys. Lett. 40, 25
(1997)
10. A. S. Villar, M. Martinelli, C. Fabre, and P. Nussenzveig,
Phys. Rev. Lett. 97, 140504 (2006)
11. S. Reynaud, A. Heidmann, E. Giacobino and C. Fabre,
Progress in Optics XXX,1 (1992)
12. D.F. Walls and G.J. Milburn, Quantum Optics, Springer
Study Edition (1995)
13. R.Graham, H.Haken, Z.Phys, 210, 276 (1968); 210, 319
(1968); 211, 469 (1968).
14. J. Laurat, T. Coudreau, N. Treps, A. Maˆ ıtre, and C.
Fabre, Phys. Rev. Lett. 91, 213601 (2003)
15. K.L.McNeil, C.W.Gardiner, Phys.Rev. A, 28(3), 1560,
(1983)
16. E. Sudarshan, Phys. Rev. Lett. 10, 277 (1963)
17. R.J. Glauber, Phys. Rev. Lett. 10, 84 (1963)
18. Yu. M. Golubev and I.V. Sokolov, Sov. Phys. JETP 60,
234 (1984)
19. K.Dechoum, P.D.Drummond, S.Chaturvedi, M.D.Reid,
”Critical fluctuations and entanglement in the nondegen-
erate parametric oscillator”
20. Yu. M. Golubev, Sov. Phys. JETP 38, 228 (1974)
21. K.V.Kheruntsyan, K.G.Petrosyan, Phys. Rev. A,62,
015801 (2000);
22. J. Laurat, T. Coudreau, N. Treps, A. Maˆ ıtre and C.
Fabre, Phys. Rev. A 69, 033808 (2004)

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T. Golubeva, Yu. Golubev, C. Fabre, N. Treps: Quantum state of an injected TROPO above threshold13
A Case of an asymmetrical injection
A.1 Master equations
In the main part of the article, we discussed in detail
how the phase locking phenomenon produced by a sym-
metrical injection on the idler and signal modes by an
external coherent light acts on the statistical properties
of the OPO radiation. In this appendix we consider the
case of an asymmetrical injection, more precisely when
the idler mode is injected by a coherent field with ampli-
tude
?Nin
in the form:
i
=
√Ninand the signal mode by the vacuum
s = 0. Let us first rewrite the master equation
state
?Nin
∂P(αp,αi,αs,t)
∂t
+κp
2
∂P
∂αs
=κi
2
∂
∂αsαsP +κs
?
∂2P
∂αi∂αs
2
∂
∂αi(αi−
√
Nin)P +
∂
∂αp(αp−
?
Nin
p)P + gαiαs∂P
∂αp
− αpα∗
s
∂P
∂αi−
−αpα∗
i
?
+ gαp
+ c.c. (137)
The corresponding classical equations read
˙ αp= −κp
˙ αi= −κi
˙ αs= −κs
2
(αp−
(αi−
αs+ gαpα∗
?
√
Nin
p) − gαiαs
Nin) + gαpα∗
(138)
2
s
(139)
2
i
(140)
Putting
κi= κs(1 − µ) (141)
one can obtain the stationary solutions of the classical
equations in the form:
αp=?Np,αi,s=
√N,(142)
where
Np=
κ2
4g2,
s
N = (µp− 1)κsκp
4g2, (143)
and
κsN = (1 − µ)κiN = (µp− 1)κpNp. (144)
In the limit of small amplitude and phase fluctuations it
is possible to factorize the solutions in its phase and am-
plitude factors:
P(αp,αi,αs,t) = P(εp,εi,εs,t)P(ϕp,ϕi,ϕs,t). (145)
We now introduce to usual sum and difference notations
ε±= εi± εs,ϕ±= ϕi± ϕs
(146)
and convert equation (137) into two equations:
∂P(εp,ε+,ε−,t)
∂t
?µκi
+κi(1 −3µ
?∂P
∂P(ϕp,ϕ+,ϕ−,t)
∂t
∂
∂ϕ+(κi(1 −3µ
−κs
4N ∂ϕ2
∂2
∂ϕ2
=
?1
2
∂
∂εp(κpεp+ κsε+)+
?
∂ε−ε−− κsN∂2
∂ε2
+
∂
∂ε+
4
ε+−(µp− 1)κp
√1 − µ
εp
+ κsN∂2
∂ε2
+
+
4)
∂
−
?
P(εp,ε+,ε−,t) +
+µ
∂t
?
1
, (147)
=
?κp
2
∂
∂ϕp(ϕp+(µp− 1)
√1 − µ
ϕ+)+
+
4) ϕ+− κsϕp) −
+µκi
4
∂2
+
∂
∂ϕ−ϕ−+
+
κs
4N
−
?
P(ϕp,ϕ+,ϕ−,t) + µ
?∂P
∂t
?
1
. (148)
Here the selected terms on the right of the equations have
the following form
?∂P
?∂P
∂t
?
?
1
=κi
4
?
?
∂
∂ε+ε−+
∂
∂ϕ+ϕ−+
∂
∂ε−ε+
∂
∂ϕ−ϕ+
?
P(εp,ε+,ε−,t), (149)
∂t
1
=κi
4
?
P(ϕp,ϕ+,ϕ−,t).
(150)
One can see that their role is essential when the synchro-
nizing parameter µ is big. Then some additional mixing
the variables εl takes place. To avoid it we require that
µ ≪ 1. A similar remark concerns the variables ϕl. In our
limit we neglect these terms, so that the Glauber quasi-
probability is factorized in the form
P(αp,αi,αs,t) = P(εp,ε+,t)P(ε−,t)P(ϕp,ϕ+,t)P(ϕ−,t)
(151)

Page 14

14 T. Golubeva, Yu. Golubev, C. Fabre, N. Treps: Quantum state of an injected TROPO above threshold
and correspondingly the equations are decoupled as
∂P(εp,ε+,t)
∂t
∂
∂ε+
+ κsN∂2
=
?1
2
∂
∂εp(κpεp+ κsε+)+
ε+−(µp− 1)κp
?
?
+
?µκi
4
√1 − µ
εp
?
+
∂ε2
+
P(εp,ε+,t),(152)
∂P(ε−,t)
∂t
=κi(1 − 3µ/4)
∂
∂ε−ε−− κsN∂2
∂ε2
−
?
P(ε−,t),
(153)
∂P(ϕp,ϕ+,t)
∂t
=
?κp
2
∂
∂ϕp(ϕp+(µp− 1)
√1 − µϕ+)+
+
∂
∂ϕ+(κi(1 − 3µ/4)ϕ+− κsϕp) −κs
4N
∂2
∂ϕ2
+
?
P(ϕp,ϕ+,t),
(154)
∂P(ϕ−,t)
∂t
=
?µκi
4
∂
∂ϕ−ϕ−+κs
4N
∂2
∂ϕ2
−
?
P(ϕ−,t). (155)
In order to analyze the time dependent (spectral) cor-
relation functions, we use the Langevin equations of the
system, which are have the following form:
˙ εp= −κp/2 εp− κs/2 ε+,
˙ ε+= −κiµ/4 ε++ κp(µp− 1)/?1 − µ εp+ f+(t),
˙ ε−= −κi(1 − 3µ/4) ε−+ f−(t),
˙ ϕp= −κp/2 ϕp− κp(µp− 1)/(2?1 − µ) ϕ+,
˙ ϕ+= −κi(1 − 3µ/4) ϕ++ κsϕp+ g+(t),
˙ ϕ−= −κiµ/4 ϕ−+ g−(t),
(156)
(157)
(158)
(159)
(160)
(161)
where the stochastic sources are determined by the pair
correlation functions
?f+(t)f+(t′)? = 2κsN δ(t − t′),
?f−(t)f−(t′)? = −2κsN δ(t − t′),
?g+(t)g+(t′)? = −κs/(2N) δ(t − t′),
?g−(t)g−(t′)? = κs/(2N) δ(t − t′).
(162)
(163)
(164)
(165)
The solution of these equations is found in the Fourier
transforms ε±,ω, εp,ω, ϕ±,ω, ϕp,ω. Their variances and cor-
relation functions are given by:
(ε2
(ε2
(εpε+)ω= −2Nκ2
(ε2
−)ω= −2N
+)ω= 2Nκs(κ2
p+ 4ω2)/Λε, (166)
p)ω= 2Np(µp− 1)κiκsκp/Λε,(167)
sκp/Λε,(168)
κs
κ2
i(1 − 3µ/4)2+ ω2,(169)
and
(ϕ2
p)ω= −
1
2NpΛϕ(µp− 1)κ2
1
2NΛϕκs(κ2
1
2NΛϕκiκ2
1
2N
iκp,(170)
(ϕ2
+)ω= −
p+ 4ω2),(171)
(ϕpϕ+)ω=
p(µp− 1)?1 − µ,
κs
(κiµ/4)2+ ω2.
(172)
(ϕ2
−)ω=(173)
where
Λϕ =
?
2ω2− κiκp[1 − 3µ/4 + (µp− 1)?1 − µ ]
+ω2[κp+ 2κi(1 − 3µ/4)]2.
Λε =
?
+ω2(κp+ κiµ/2)2
?2
+
(174)
2ω2− κpκi[µ/4 + (µp− 1)?1 − µ ]
?2
+
(175)

Page 15

T. Golubeva, Yu. Golubev, C. Fabre, N. Treps: Quantum state of an injected TROPO above threshold15
Fig. 1. Purity of the output field as a function of the dimen-
sionless frequency for synchronizing field parameter a) µ = 0.1;
b) µ = 0.35 and different excesses above threshold for symme-
try phase locking. Both positive and negative frequency do-
mains are plotted, however only the positive one correspond to
physical values
Fig. 2. Partial purity of the two-mode OPO compared to the
system purity (including the pump mode). One note that the
partial purity is always larger the total one.