Tania Golubeva

Yuri Golubev

Claude Fabre

Nicolas Treps

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

Keywords: 4 2, 50, Dv, 42, 50, Lc

may come

Introduction

Three-wave nonlinear interaction in a χ 2 medium 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 addition, in the bellow threshold, one has generally the possibility to consider the pump as a coherent one with a fixed classical amplitude and thus to restrict the analysis 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 quantum modes are degenerate [1], and twin, quantum intensity 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 interesting 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 significantly 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 between 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 correlations 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 concerning states produced by conditional measurements performed 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 moment 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 propagating from the generating optical device to the detectors, even though such a concept is often used in the commu-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 perform 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 "toymodel" 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 [START_REF] Walls | Quantum Optics[END_REF]11] is undoubtedly valid except very close to the oscillation threshold. As is well-known, as far as the signalidler phase difference is concerned, a phase diffusion phenomenon takes places, giving rise to diverging phase difference fluctuations at very long times [START_REF] Graham | [END_REF]. 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 covariance matrix for the spectral components of the quantum fluctuations of the three output modes. This enables us to calculate the spectral purity of the TROPO quantum 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 quantity 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 fluctuations of the three interacting modes. In the next section, we determine and discuss the purity of the system from the expression of the covariance matrix. We then derive 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.

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-conversion process ω p → ω i +ω s 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 [START_REF] Graham | [END_REF]15]:

V = i g âp â † i â † s -h.c. (1)
âl

, â † l = 1, l = p, i, s
In this approach, the three interacting intra-cavity modes are quantized. The master equation in the interaction picture for three-mode field density matrix ρ is :

ρ = - i V , ρ -Rρ + m=p,i,s Dm ρ. (2)
The operator R 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:

Rρ = m=p,i,s κ m 2 â † m âm ρ + ρâ † m âm -2â m ρâ † m . ( 3
)
where κ m is the energy damping rate of the m-mode. The operators Dm ensure an excitation of each of the actual modes by a quasi-classical or coherent state of amplitude

N in m : Dm ρ = κ m 2 N in m â † m -âm , ρ (4)
The external quasi-classical fields with a real amplitudes N in m are in resonance with the m-waves. The excitation of the idler and signal waves by the external coherent waves is used to depress the phase diffusion in the system. For this aim, it would be enough to put N in i ≡ √ N in = 0 and N in s = 0. However the asymmetry arising here complicates both the mathematical and physical situations. Much simpler solutions take place as

N in i = N in s ≡ √ N in .
In this article we shall consider both models focusing in the body of the article 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 for the three interacting modes. The Glauber function P is introduced by the integral relation:

ρ(t) = P (α p , α i , α s , t) |α p , α i , α s × α p , α i , α s |d 2 α p d 2 α i d 2 α s . (5)
With symmetrical phase locking, the master equation reads:

∂P (α p , α i , α s , t) ∂t = m=i,s κ m 2 ∂ ∂α m (α m - √ N in )P + + κ p 2 ∂ ∂α p (α p -N in p )P + g(α i α s ∂P ∂α p -α p α * s ∂P ∂α i - -α p α * i ∂P ∂α s ) + gα p ∂ 2 P ∂α i ∂α s + 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 remains 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 momenta 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.

Classical equations for the OPO operation

As is well-known, to get the semi-classical evolution equations for the mean amplitudes, we can use Eq (6), neglecting the second derivatives with respect to the complex amplitude. Then one can get:

αp = - κ p 2 (α p -N in p ) -gα i α s (7) αi = - κ i 2 (α i - √ N in ) + gα p α * s (8) αs = - κ s 2 (α s - √ N in ) + gα p α * i (9)
In the following we will only consider the case of equal losses for the signal and idler cavity losses:

κ i = κ s ≡ κ. ( 10
)
Then it is not difficult to obtain the stationary solutions, which can be written in the form:

α p = N p , α i = α s = √ N , (11)
where for the values N and N p the following equalities take place:

g N p = κ 2 (1 -µ), gN N p = κ p 2 (µ p -1), κ(1 -µ)N = κ p (µ p -1)N p . (12)
These equations depend on two dimensionless parameters µ p and µ. µ p , the pump parameter, is defined as

µ p = N in p N th (13)
where N th = κ 2 (1µ)/(4g 2 ). µ p > 1 corresponds to the above threshold regime. µ, the injection parameter, represents which fraction of the total signal and/or idler fields is injected :

µ = N in 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 Poissonian statistics of the external field would be imposed on the intracavity mode and its quantum properties would be destroyed.

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

α m = √ u m e iϕ m , m = p, i, s, (15)
then we can require

u m = N m + ε m , ε m ≪ N m . (16)
Secondly, because of injection, one can find well defined values for the steady state phases, namely ϕ m = 0. We will assume that in the system the phase fluctuations around these steady state values are small :

ϕ m ≪ 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 (α p , α i , α s , t) = (18) = P (ε p , ε + , t)P (ε -, t)P (ϕ p , ϕ + , t)P (ϕ -, t)
with decoupled equations for the different factors:

∂P (ε p , ε + , t) ∂t = 1 2 ∂ ∂ε p (κ p ε p + κ(1 -µ)ε + )+ + ∂ ∂ε + µκ 2 ε + -κ p (µ p -1)ε p + +κN (1 -µ) ∂ 2 ∂ε 2 + P (ε p , ε + , t), (19)
∂P (ε -, t) ∂t = κ(1 -µ/2) ∂ ∂ε - ε -- -κN (1 -µ) ∂ 2 ∂ε 2 - P (ε -, t), (20)
∂P (ϕ p , ϕ + , t) ∂t = κ p 2 ∂ ∂ϕ p (ϕ p + (µ p -1)ϕ + )+ + ∂ ∂ϕ + (κ(1 -µ/2)ϕ + -κ(1 -µ)ϕ p ) - - κ 4N (1 -µ) ∂ 2 ∂ϕ 2 + P ( ϕ p , ϕ + , t), (21)
∂P (ϕ -, t) ∂t = µκ 2 ∂ ∂ϕ - ϕ -+ κ 4N (1 -µ) ∂ 2 ∂ϕ 2 - P (ϕ -, t). (22)
Here

ε ± = ε i ± ε s , ϕ ± = ϕ i ± ϕ s (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 synchronizing fields influence the noise properties of the system. It is important to understand whether the phase locking 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.

Intracavity spectral densities

In order to analyze the time dependent correlation function, let us derive the Langevin equations for the three interacting fields ( 19)- (22). They are easily written according to well-known rules and have the following form:

εp = -κ p /2 ε p -κ/2(1 -µ) ε + , (24) ε+ = -κµ/2 ε + + κ p (µ p -1) ε p + f + (t), (25) ε-= -κ(1 -µ/2) ε -+ f -(t), ( 26
) φp = -κ p /2 ϕ p -κ p /2(µ p -1) ϕ + , ( 27
) φ+ = -κ(1 -µ/2) ϕ + + κ(1 -µ) ϕ p + g + (t), (28) φ-= -κµ/2 ϕ -+ g -(t), (29)
where the stochastic sources are determined by the pair correlation functions

f + (t)f + (t ′ ) = 2κN (1 -µ) δ(t -t ′ ), (30) f -(t)f -(t ′ ) = -2κN (1 -µ) δ(t -t ′ ), (31) g + (t)g + (t ′ ) = -κ(1 -µ)/(2N ) δ(t -t ′ ), (32) g -(t)g -(t ′ ) = κ(1 -µ)/(2N ) δ(t -t ′ ). ( 33
)
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

(ε 2 + ) ω = 2N × (34) × κ(κ 2 p + 4ω 2 )(1 -µ) [2ω 2 -κ p κ[µ/2 + (µ p -1)(1 -µ)]] 2 + ω 2 (κ p + κµ) 2 , (ε 2 p ) ω = 2N p × (35) × κ 2 κ p (1 -µ) 2 (µ p -1) [2ω 2 -κ p κ[µ/2 + (µ p -1)(1 -µ)]] 2 + ω 2 (κ p + κµ) 2 , (ε p ε + ) ω = -2N × (36) × κ 2 κ p (1 -µ) [2ω 2 -κ p κ[µ/2 + (µ p -1)(1 -µ)]] 2 + ω 2 (κ p + κµ) 2 , (ε 2 -) ω = -2N κ(1 -µ) κ 2 (1 -µ/2) 2 + ω 2 , (37)
and

(ϕ 2 p ) ω = - 1 2N p × (38) × κ 2 κ p (µ p -1)(1 -µ) 2 [2ω 2 -κκ p [µ/2 + µ p (1 -µ)]] 2 + ω 2 [κ p + 2κ(1 -µ/2)] 2 , (ϕ 2 + ) ω = - 1 2N × (39) × κ(κ 2 p + 4ω 2 )(1 -µ) [2ω 2 -κκ p [µ/2 + µ p (1 -µ)]] 2 + ω 2 [κ p + 2κ(1 -µ/2)] 2 , (ϕ p ϕ + ) ω = 1 2N × (40) × κκ 2 p (µ p -1)(1 -µ) [2ω 2 -κκ p [µ/2 + µ p (1 -µ)]] 2 + ω 2 [κ p + 2κ(1 -µ/2)] 2 , (ϕ 2 -) ω = 1 2N κ(1 -µ) (κµ/2) 2 + ω 2 . ( 41
)
In these expressions, the spectral density (ε 2 m ) ω and (ϕ 2 m ) ω are defined as a factor in front of the delta-functions in the correlation functions

ε m (ω) ε m (ω ′ ) = (ε 2 m ) ω δ(ω + ω ′ ), m = p, ±, ϕ m (ω) ϕ m (ω ′ ) = (ϕ 2 m ) ω δ(ω + ω ′ ), (42)
where

ε m (ω) = 1 √ 2π ε m (t) e -iωt dt, ε m (t) = 1 √ 2π ε m (ω) e iωt dω. (43)
The mutual correlations (ε p ε + ) ω and (ϕ p ϕ + ) ω are defined exactly in the same way:

ε + (ω) ε p (ω ′ ) = (ε + ε p ) ω δ(ω + ω ′ ), ϕ + (ω) ϕ p (ω ′ ) = (ϕ + ϕ p ) ω δ(ω + ω ′ ). ( 44
)
3 Spectral purity of the OPO quantum state

Output field variances and correlations

In the previous section, we have considered the intracavity spectral densities. Now we want to determine the corresponding 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 âm (t) (m = p, i, s) that obey the commutation relations

âm (t), â † m (t) = δ mn . ( 45
)
Let us call Âm (t) the corresponding operator for the output beams. They obey the commutation relations:

Âm (t), Ân (t ′ ) † = δ mn δ(t -t ′ ), Âm (t), Ân (t ′ ) = 0. ( 46
)
and are related to the intracavity ones by the input-output relations on the coupling mirror:

Âm (t) = √ κ m âm (t) -C m + Âm,vac (t) , m = p, i, s. (47)
We have taken into account the fact that the reflection coefficient of the output mirror is about one. C p is the complex normalized amplitude of the pump wave in resonance with the p-mode. This amplitude is related to the quantity N in p introduced earlier by relation C p = κ p N in p /2. The amplitudes C s and C i are the classical injected fields in resonance with the idler and signal modes:

C i = C s = √ κN in /2.
Âm,vac (t) are the input vacuum fluctuations, with commutation relations Âm,vac (t), Ân,vac (t

′ ) † = δ mn δ(t -t ′ ), Âm,vac (t), Ân,vac (t ′ ) = 0. ( 48
)
We can rewrite (47) for the fluctuations δ Âm = Âm -Âm and δâ m = âmâm , in the Fourier domain:

δ Âm (ω) = √ κ m δâ m (ω) -Âm,vac (ω). ( 49
)
Let us divide the frequency scale into small equal intervals of size ∆. Then for each discrete frequency ω l one can write:

δ Âl m = 1 √ ∆ ω l +∆/2 ω l -∆/2 δ Âm (ω) dω. ( 50
)
It is easy to check that the algebra of these operators is determined by the commutation relations:

δ Âl m , ( Âk m ) † = δ lk . (51)
Thus the operators δ Âl m , ( Âl m ) † can be thought of as being the annihilation and creation operators for the photons in the frequency band around ω l . Then for each mode (frequency) one can introduce the corresponding quadrature components via the standard relations

δ Xl m = 1 2 (δ Âl m ) † + δ Âl m , δ Ŷ l m = i 2 (δ Âl m ) † -δ Âl m . (52)
Taking into account (49) the variances of the output quadratures can then be expressed in terms of the intracavity ones:

δ X2 m l = 1 4 + κ m 4N m 1 ∆ ω l +∆/2 ω l -∆/2 (ε 2 m ) ω dω, δ Ŷ 2 m l = 1 4 + κ m N m 1 ∆ ω l +∆/2 ω l -∆/2 (ϕ 2 m ) ω dω. ( 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(δ X2 m ) ω = 1 + κ m /N m (ε 2 m ) ω , 4(δ Ŷ 2 m ) ω = 1 + 4κ m N m (ϕ 2 m ) ω . (54) 4(δ Xm δ Xn ) ω = κ m /N m κ n /N n (ε m ε n ) ω , 4(δ Ŷm δ Ŷn ) ω = 4 κ m N m κ n N n (ϕ m ϕ n ) ω , m, n = p, i, s. (55)

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:

M ω = M ǫ 0 0 M φ , (56)
where ||M ǫ || are amplitude and ||M φ || are phase 3 × 3 matrices given by:

M ǫ =   4(δX 2 p ) ω 4(δX p δX i ) ω 4(δX p δX s ) ω 4(δX i δX p ) ω 4(δX 2 i ) ω 4(δX i δX s ) ω 4(δX s δX p ) ω 4(δX s δX i ) ω 4(δX 2 s ) ω   ,( 57
)
and analogically ||M φ || with help of the variances 4(δY m δY n ) ω . The spectral variances are determined as a factor in front of the delta-function in the correlation relation:

δX m (ω)δX n (ω ′ ) = (δX m δX n ) ω δ(ω + ω ′ ), δY m (ω)δY n (ω ′ ) = (δY m δY n ) ω δ(ω + ω ′ ) (58)
A widely used way of describing the properties of a quantum 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 ρω , that can be experimentally characterized for example by quantum tomography. Let us introduce the spectral purity, given by

Π ω = T r ρ2 ω . ( 59
)
This state is a pure state when Π ω = 1, or a statistical mixture when Π ω < 1. In the present case of small fluctuations and input Gaussian states, all quantum fluctuations have Gaussian statistics. Then the purity is equal to

Π ω = 1 √ det M ω . ( 60
)

Spectral purity for a symmetrical injection

In this simple case, we will use the basis of sum and difference 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:

det M ω = N -D X D Y , (61)
where

N -= 2 δ X2 -ω 2 δ Ŷ 2 -ω , D X = 2 δ X2 + ω 4 δ X2 p ω -4 δ X+ δ Xp 2 ω , D Y = 2 δ Ŷ 2 + ω 4 δ Ŷ 2 p ω -4 δ Ŷ+ δ Ŷp 2 ω , (62)
One can see that N -is related to the uncertainty relation for the differential variances. D X,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 example and for the simplicity of the calculation, the case κ(µ p -1), κ ≪ κ p . According to the previous subsection the differential variances are derived in the form

2 δ X2 -ω = 1 + κ 2N (ε 2 -) ω = = 1 - κ 2 (1 -µ) κ 2 (1 -µ/2) 2 + ω 2 , ( 63
) 2 δ Ŷ 2 -ω = 1 + 2κN (ϕ 2 -) ω = 1 + κ 2 (1 -µ) (κµ/2) 2 + ω 2 . (64)
From these expressions, one retrieves first a striking but well-known result, characteristic of the non-degenerate OPO above threshold: N -= 1 for any values of the parameters such as the frequency, µ p , and µ.

In order to get the other determinants D X,Y we need to derive the corresponding variances in the explicit form:

2 δ X2 + ω = 1 + κ 2N (ε 2 + ) ω = = 1 + κ 2 (1 -µ) κ 2 [µ/2 + (1 -µ)(µ p -1)] 2 + ω 2 ,(65) 2 δ Ŷ 2 + ω = 1 + 2κN (ϕ 2 + ) ω = = 1 - κ 2 (1 -µ) κ 2 [µ/2 + (1 -µ)µ p ] 2 + ω 2 , ( 66
) 4 δ X2 p ω = 1 + κ p N p (ε 2 p ) ω = = 1 + 2κ 2 (µ p -1)(1 -µ) 2 κ 2 [µ/2 + (1 -µ)(µ p -1)] 2 + ω 2 ,(67) 4 δ Ŷ 2 p ω = 1 + 4κ p N p (ϕ 2 p ) ω = = 1 - 2κ 2 (1 -µ) 2 (µ p -1) κ 2 [µ/2 + (1 -µ)µ p ] 2 + ω 2 , ( 68
) 2 δ Xp δ X+ ω = κ p 2N p κ 2N (ε p ε + ) ω = = - κ 2 (1 -µ)(µ p -1) κ 2 [µ/2 + (µ p -1)(1 -µ)] 2 + ω 2 , ( 69
) 2 δ Ŷp δ Ŷ+ ω = 2κ p N p √ 2κN (ϕ p ϕ + ) ω = = κ 2 (µ p -1)(1 -µ) 3 κ 2 [µ/2 + µ p (1 -µ)] 2 + ω 2 . ( 70
)
Let us remark that, in contrast to N -, the quantities

N + = 2 δ X2 + ω 2 δ Ŷ 2 + ω , N p = 4 δ X2 p ω 4 δ Ŷ 2 p ω (71)
are non minimum, especially near zero frequencies when the pump is not strongly above threshold. For example, when µ p -1 ≫ µ and ω = 0

N + = N p = (µ p -1) 2 + 1 µ 2 p µ p + 1 µ p -1 (72)
N + and N p take 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 (µ p = 4, 2 and 1,1). One first observes that the spectral purity is equal to one outside 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 (µ p = 1, 1).

We are therefore led to the conclusion that, close to threshold, the three-mode "single noise frequency state" describing the OPO is mixed at low noise frequencies. Considering that we have taken into account in a quantum way all the interacting modes, and that all the intracavity 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 redistribution 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 twomode 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 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

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 κ i = κ s are excited equally by the external fields with amplitudes

N in i = N in s .
Here we want to briefly discuss the role of the asymmetry in the injection. The corresponding derivations can be found in the appendix of this paper, where the physical situation with N in i = N in and N in s = 0 is considered. In this case, the mean output signal and idler fields are no longer equal. In order to retrieve some symmetry we further require that κ s = κ i (1µ), so that the equality N i = N s ≡ 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 beam and the intracavity variances turns out to be more complicated for the ±-variances:

2 δ X2 ± ω = 1 + κ i 8N [(ε 2 ± ) ω (1 + 1 -µ) 2 + +(ε 2 ∓ ) ω (1 -1 -µ) 2 ], ( 73
) 2 δ Ŷ 2 ± ω = 1 + κ i N/2[(ϕ 2 ± ) ω (1 + 1 -µ) 2 + +(ϕ 2 ∓ ) ω (1 -1 -µ) 2 ]. (74)
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 spectral band κ the purity is close to 1, as in the symmetrical case. The band where purity Π ω = 1, depends on the distance to threshold and decreases when the pump power increases. Well above threshold (µ p > 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 injection field even for very small values µ. In the limit of small µ and µ p > 5 we get Π ω=0 = 1/2.

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 injection. 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 δ X2 i,s ω = δ X2 + ω + δ X2 -ω (75) one has 4 δ X2 i,s ω = = 1 + 1 2 κ 2 κ 2 (µ p -1 + µ/4) 2 + ω 2 - 1 2 κ 2 κ 2 + ω 2 . (76)
Strongly above threshold µ p ≫ 1 the second term turns out to be negligible and the maximum squeezing is reached in the idler and signal waves.

4 δ X2 i,s ω=0 = 1 2 . ( 77
)
We can expect squeezing for µ p > 2.

In contrast with the amplitude squeezing the phase squeezing is found only in the pump mode:

4 δ Ŷ 2 p ω = 1 - 2κ 2 (µ p -1) κ 2 µ 2 p + ω 2 . ( 78
)
On the zero frequency

4 δ Ŷ 2 p ω=0 = (µ p -1) 2 + 1 µ 2 p . (79)
It is not difficult to see that maximum squeezing 4 δ Ŷ 2 p ω=0 = 1/2 is reached as µ p = 2.

In order to evaluate an entanglement of the idler and signal waves, we use the Duan criterium

2 δ X2 -, 2 δ Ŷ 2 + < 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 (µ p ≫ 1) 2 δ Ŷ 2 +ω=0 → 1.

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 components, by solving non stationary equations in Fourier space. Stationary master equations also contain important pieces of information about the quantum system. We devote 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 fluctuations 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 subsections the solutions of the amplitude and phase equations respectively.

Amplitude quasi-probability distribution

The stationary amplitude quasi-probability P (ε p , ε i , ε s ) can be factorized in the form

P (ε p , ε i , ε s ) = P (ε p , ε + )P (ε -), ( 81
)
where each of the factors obeys its own equations:

κ(1 -µ/2) ∂ ∂ε - ε --κN (1 -µ) ∂ 2 ∂ε 2 - P (ε -) = 0, (82) 1 2 ∂ ∂ε p (κ p ε p + κ(1 -µ)ε + ) + ∂ ∂ε + (κµ/2 ε + - -κ p (µ p -1)ε p ) + κN (1 -µ) ∂ 2 ∂ε 2 + P (ε p , ε + ) = 0. ( 83
)
Let us first consider the second equation. The solution has the Gaussian form :

P (ε p , ε + ) = 1 2π D p D + exp - (ε p -aε + ) 2 2D p - ε 2 + 2D + . (84)
The unknown parameters D p , D + and a are coupled with the variances by means of the following relations :

D + = ε 2 + , D p = ε 2 p - ε + ε p 2 ε 2 + , a = ε + ε p ε 2 + . (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 D p , D + and a. From eq. ( 83) one can then obtain the algebraic system of equation for the variances in the form :

-κ p ε 2 p -κ(1 -µ) ε p ε + = 0, ( 86
) - 1 2 (κ p + κµ) ε p ε + - 1 2 κ(1 -µ) ε 2 + + κ p (µ p -1) ε 2 p = 0, ( 87
)
-κµ ε 2 + + 2κ p (µ p -1) ε p ε + + 2κN (1 -µ) = 0, ( 88
)
Solving this system, one can get the variances :

ε 2 + = N 1 µ/2 + µ p -1 1 + 2κ κ p + κµ (µ p -1) , ( 89
)
ε + ε p = -N p 1 µ/2 + µ p -1 κ p κ p + κµ (µ p -1), ( 90
)
ε 2 p = N p 1 µ/2 + µ p -1 κ κ p + κµ (µ p -1). ( 91
)
In the limit κ, κ(µ p -1) ≪ κ p and µ ≪ 1 these variances read

ε 2 + = N 1 µ/2 + µ p -1 , ε 2 p = N p µ p -1 µ/2 + µ p -1 κ κ p , (92)
ε + ε p = -N p µ p -1 µ/2 + µ p -1 .
The knowledge of the variances provides us with a possibility to find the parameters of the Gaussian quasiprobability:

D + = N 1 µ/2 + µ p -1 , D p = N p µ p -1 µ/2 + µ p -1 µκ κ p , ( 93
) a = - κ κ p .
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 connected with the minus sign in front of the second derivative with respect to ε -. On the one hand this minus sign informs 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 ε 2 -turns out to be negative

ε 2 -= -N (94)
although the following relation

ε 2k -= (2k -1)!! ε 2 - k , (95)
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 k d 2k dε 2k - δ(ε -). ( 96
)
It is also possible to use another formal equivalent expression:

P (ε -) = exp - N 2
d 2 dε 2 - δ(ε -) (97)
From these expressions, one can derive Fano parameters for each of the intracavity modes:

F i,s = 1 4 2 -µ p µ/2 + µ p -1 + 1, F p = 1. ( 98
)

Phase quasi-probability distribution

As for the amplitude, the quasi-probability distribution of the three phases can be factorized in the form:

P (ϕ p , ϕ i , ϕ s ) = P (ϕ p , ϕ + )P (ϕ -). ( 99
)
The corresponding master equation read ( 19)-( 22):

κ p 2 ∂ ∂ϕ p (ϕ p + (µ p -1)ϕ + ) + ∂ ∂ϕ + (κ(1 -µ/2)ϕ + - -κ(1 -µ)ϕ p ) - κ 4N (1 -µ) ∂ 2 ∂ϕ 2 + P ( ϕ p , ϕ + ), ( 100
)
µκ 2 ∂ ∂ϕ - ϕ -+ κ 4N (1 -µ) ∂ 2 ∂ϕ 2 - P (ϕ -). ( 101
)
Again the minus sign in front of the last term means quantum phase features in the field. A formal solution of the equation can be presented in the form of the distribution:

P (ϕ p , ϕ + ) = ∞ m,n=0 (-1) m+n m! n! M mn d m dϕ m p d n dϕ n + δ(ϕ p ) δ(ϕ + ), M mn = ϕ m p ϕ n + (102)
It is possible to demonstrate that the momenta M mn are 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 κ p /2 + n κ)M mn -m (µ p -1)κ p /2 M m-1n+1 + +nκM m+1n-1 -n(n -1)κ/(4N ) M mn-2 = 0. ( 103
)
In particular, as κ,

(µ p -1)κ ≪ κ p ϕ 2 p = - 1 4N p µ p -1 µ p κ κ p , ϕ 2 + = - 1 4N 1 µ p , ( 104
)
ϕ p ϕ + = 1 4N p 1 µ p κ κ p .
As for Eq (101) it describes a stationary Gaussian process and its solution is well-known:

P (ϕ -) = 1 π/(µN ) exp - ϕ 2 - 1/(µN ) (105)

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 oscillators with different frequencies. In contrast, the P-function that we have just derived describes the system in its stationary state, but only for the three intracavity oscillators. 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 singleoscillator 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 l 0 = cτ , where τ the correlation time of the intracavity field. Then it is formally possible to introduce photon operators âm and â † m acting in m-th layer such as âm , â † n = δ mn . This means that the field outside the cavity can be presented as a set of spatially located 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 fluctuations over time scales much smaller than τ . One can then define a quasi-probability distribution P out for such output oscillators, which is related to the already calculated intracavity P-function by means of

P out (α p,out , α i,out , α s,out ) = = 1 T p T i T s P α p,out T p , α i,out √ T i , α s,out √ T s . (106)
√ T i = √ T s ≡ √
T being the amplitude transmission coefficients of the coupling mirror for the signal and idler modes;

T p the transmission coefficient for the pump mode; α p,out , α i,out α s,out are the corresponding complex amplitudes, which are the eigenvalues of the annihilation photon operator.

Knowing P out of 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:

Π st = Π 1 × Π 2 × Π 3 × Π 4 , (107)
where the amplitude factors are

Π 1 = dε p dε + P out (ε p , ε + ) dε ′ p dε ′ + P out (ε ′ p , ε ′ + ) × × exp - (ε p -ε ′ p ) 2 4N p T p exp - (ε + -ε ′ + ) 2 8N T , (108)
Π 2 = dε -P out (ε -) dε ′ -P out (ε ′ -) × × exp - (ε --ε ′ -) 2 8N T , (109)
and the phase ones are:

Π 3 = dϕ p dϕ + P out (ϕ p , ϕ + ) dϕ ′ p dϕ ′ + P out (ϕ ′ + , ϕ ′ p )× × exp - (ϕ p -ϕ ′ p ) 2 1/(N p T p ) exp - (ϕ + -ϕ ′ + ) 2 2/(N T ) , (110)
Π 4 = dϕ -P out (ϕ -) dϕ ′ -P out (ϕ ′ -) × × exp - (ϕ --ϕ ′ -) 2 2/(N T ) . (111)
Substituting Eqs. ( 84), ( 96), (102), and (105) and considering the limit κ, (µ p -1)κ ≪ κ p we get

Π 1 = 1 (1 + ν T /µ) 1/2 , Π 2 = Π 3 = 1, (112)
Π 4 = 1 (1 + T /µ) 1/2 , where ν = µ(µ p -1/2) µ/2 + µ p -1 , (113)
If the output mirror of the cavity is highly reflecting, so that that T ≪ µ, then

Π st = 1. (114)
A similar calculation performed with the intracavity Pfunction with the same parameters gives Π 1 = Π 2 = Π 3 = 1, and Π 4 = √ µ, so that the purity of the intracavity state, Π = µ 1/2 is much smaller than 1.

Field fluctuations integrated over short time intervals depend only on the high frequency fluctuations of the output fields. The property that Π st = 1 is therefore certainly connected to the already noticed feature that Π ω = 1 for noise frequencies larger than the cavity bandwidth.

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 quantum 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 C in (n i , n s ) to find n i photons in the idler intracavity mode and simultaneously n s photons in the signal intracavity mode is defined as

C in (n i , n s ) = np=0,1,... n p n i n s |ρ|n p n s n i , ( 115
)
where ρ is the stationary three-mode intracavity density matrix. By using the Glauber diagonal representation (7) in (78), we have

C in (n i , n s ) = 1 4 ∞ 0 ∞ 0 du i du s P red. (u i , u s ) × ×e -ui u ni i n i ! e -us u ns s n s ! , (116)
where P red. is a "reduced"Glauber photon quasi-probability given by

P red. (u i , u s ) = d 2 α p dϕ i dϕ s P (α p , α i , α s ), ( 117
)
where α l = √ u l exp(iϕ l )(l = s, i). From the previous section

P red. (u i , u s ) = P (ε -) dε p P (ε p , ε + ) = (118) = 1 2πD + exp - ε 2 + 2D + exp - N 2
d 2 dε 2 - δ(ε -)
Substituting this into (116) one can get after the corresponding integrating one can obtain that

C in (n i , n s ) = 1/ √ 2πλN exp - (n + -2N ) 2 2λN × ×1/ √ 2πN exp - n 2 - 2N , (119)
where

λ = 2 + 1 µ/4 + µ p -1 (120) (κ, (µ p -1)κ ≪ κ p ).
This function describes the probabilities to find n i photons in the idler mode and n s photons in the signal mode inside 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 intracavity fields, but with its own parameters depending, in particular, on observation time τ . We will therefore write it as:

C out (n i , n s ) = 1/ 2πd + exp - (n + -2N out ) 2 2d + × ×1/ 2d -exp - n 2 - 2d - . ( 121
)
where the parameters N out , 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 ε i or ε s we obtain the single-mode photon number probability for both the idler and signal modes:

C in (n l ) = 1 √ 2πN F in exp - (n l -N ) 2 2N F in , (122)
F in = 1 4 (1 + λ), l = i, s,
and correspondingly for the output beam:

C out (n l ) = 1 √ 2πF out exp - (n l -N out ) 2 2N F out . (123)
In the last probability n l are eigen-numbers of the operator

nl (t) = t+τ /2 t-τ /2 Â † l (t ′ ) Âl (t ′ ) dt ′ , (124)
where the operators Âl and  † l describe light propagating in free space (see (46)). Then

N out = t+τ /2 t-τ /2 Â † l (t ′ ) Âl (t ′ ) dt ′ = κτ N. (125)
In order to determine the Fano-factor F out , we have to calculate the variance

n2 l -nl 2 = N out F out . (126)
For the stationary flux

N out F out = (127) = τ /2 -τ /2 τ /2 -τ /2 dt 1 dt 2 Â † l (t 1 ) Âl (t 1 ) Â † l (t 2 ) Âl (t 2 ) -N 2 out .
Following to standard procedure, one obtains

F out = 1 + κ/N dω (ε 2 l ) ω δ τ (ω), (128)
where the integral contains a product of two functions. One of them is

κ/N (ε 2 l ) ω = 1 2 κ 2 (µ/2 + µ p -1) 2 κ 2 + ω 2 - κ 2 κ 2 + ω 2 . ( 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

δ τ (ω) = 1 π sin 2 ωτ /2 τ ω 2 /2 . ( 130
)
For long enough integration times τ , δ τ (ω) gets close to a δ-function when τ → ∞, so that F out is given by

F out = 1 2 µ p µ/2 + µ p -1 . (131)
Near threshold F out = 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 F out = 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 twomode variances d ± (121): 132)

d ± = 2N out 1 + κ/(2N ) dω (ε 2 ± ) ω δ τ (ω) .(
As (µ p -1)κ, κ ≪ κ p κ/(2N ) (ε 2 + ) ω = κ 2 (µ/4 + µ p -1) 2 κ 2 + ω 2 , κ/(2N ) (ε 2 -) ω = - κ 2 κ 2 + ω 2 (133)
Inserting these expressions to formula (132), one obtains the full joint photon number probability distribution C out (n i , n s ) 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 + µ p -1) ≪ 1 the variances read:

d ± = 2N out . ( 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 + µ p -1) ≫ 1,

d + = µ p /(µ/2 + µ p -1), d -= 0. ( 135
)
We have a possibility to derive the wished photon number probability in form:

C out (n i , n s ) = (136) = 1 2πd + exp - (2n i + -2N out ) 2 2d + δ(n i -n s ).
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.

Conclusion

One of our aims here was to investigate the quantum state purity of the TROPO radiation. There is a widelydistributed opinion that the purity of the output emission state must be equal to one. The reason of that is connected 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 theories. 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 selected frequency is not one (especially near zero frequencies). 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 under the asymmetrical synchronization the purity becomes 1/2 even if µ p ≫ 1. So we can conclude that, generally speaking, the purity for the oscillator on the selected frequency is not one and essentially depends on the power of the synchronizing field and the asymmetry of the synchronization too.

We needed to introduce the synchronization into our the OPO system to depress the diffusion of the differential 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 diffusion 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, nevertheless not only the phase fluctuations are stabilized under an influence of the external field but the amplitude ones too. We have found that this phenomenon turns out to be essential near threshold as µ p -1 < µ. One can see that all amplitude variances on the zero frequency are proportional to 1/µ 2 (not to infinity as under the phase diffusion). This apiaries as well as in the spectral and stationary variances.

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 symmetrical 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 amplitude N in i = √ N in and the signal mode by the vacuum state N in s = 0. Let us first rewrite the master equation in the form:

∂P (α p , α i , α s , t) ∂t = κ i 2 ∂ ∂α s α s P + κ s 2 ∂ ∂α i (α i - √ N in )P + + κ p 2 ∂ ∂α p (α p -N in p )P + g α i α s ∂P ∂α p -α p α * s ∂P ∂α i - -α p α * i ∂P ∂α s + gα p ∂ 2 P ∂α i ∂α s + c.c. (137)
The corresponding classical equations read

αp = - κ p 2 (α p -N in p ) -gα i α s (138) αi = - κ i 2 (α i - √ N in ) + gα p α * s (139) αs = - κ s 2 α s + gα p α * i (140)
Putting

κ i = κ s (1 -µ) (141)
one can obtain the stationary solutions of the classical equations in the form:

α p = N p , α i,s = √ N , (142)
where

N p = κ 2 s 4g 2 , N = (µ p -1) κ s κ p 4g 2 , (143)
and

κ s N = (1 -µ)κ i N = (µ p -1)κ p N p . (144)
In the limit of small amplitude and phase fluctuations it is possible to factorize the solutions in its phase and amplitude factors: P (α p , α i , α s , t) = P (ε p , ε i , ε s , t)P (ϕ p , ϕ i , ϕ s , t). (145)

We now introduce to usual sum and difference notations

ε ± = ε i ± ε s , ϕ ± = ϕ i ± ϕ s (
One can see that their role is essential when the synchronizing parameter µ is big. Then some additional mixing the variables ε l takes place. To avoid it we require that µ ≪ 1. A similar remark concerns the variables ϕ l . In our limit we neglect these terms, so that the Glauber quasiprobability is factorized in the form P (α p , α i , α s , t) = P (ε p , ε + , t)P (ε -, t)P (ϕ p , ϕ + , t)P (ϕ -, t)

where the stochastic sources are determined by the pair correlation functions

f + (t)f + (t ′ ) = 2κ s N δ(t -t ′ ), (162) f -(t)f -(t ′ ) = -2κ s N δ(t -t ′ ), ( 163
) g + (t)g + (t ′ ) = -κ s /(2N ) δ(t -t ′ ), (164) g -(t)g -(t ′ ) = κ s /(2N ) δ(t -t ′ ). ( 165
)
The solution of these equations is found in the Fourier transforms ε ±,ω , ε p,ω , ϕ ±,ω , ϕ p,ω . Their variances and correlation functions are given by:

(ε 2 + ) ω = 2N κ s (κ 2 p + 4ω 2 )/Λ ε , (166)

  (155) In order to analyze the time dependent (spectral) correlation functions, we use the Langevin equations of the system, which are have the following form:εp = -κ p /2 ε pκ s /2 ε + ,(156) ε+= -κ i µ/4 ε + + κ p (µ p -1)/ 1µ ε p + f + (t), (157) ε-= -κ i (1 -3µ/4) ε -+ f -(t),(158) φp= -κ p /2 ϕ pκ p (µ p -1)/(2 1µ) ϕ + , (159) φ+ = -κ i (1 -3µ/4) ϕ + + κ s ϕ p + g + (t),(160)φ-= -κ i µ/4 ϕ -+ g -(t),

Fig. 1 .Fig. 2 .

 12 Fig. 1. Purity of the output field as a function of the dimensionless frequency for synchronizing field parameter a) µ = 0.1; b) µ = 0.35 and different excesses above threshold for symmetry phase locking. Both positive and negative frequency domains are plotted, however only the positive one correspond to physical values

Fig. 3 .

 3 Fig. 3. Purity of the output field in dependence on dimensionless frequency for synchronizing field parameter a) µ = 0.1; b) µ = 0.01 and different excesses above threshold for asymmetry phase locking

  = 2ω 2κ i κ p [1 -3µ/4 + (µ p -1) 1µ ] [κ p + 2κ i (1 -3µ/4)] 2 . (174)Λ ε = 2ω 2κ p κ i [µ/4 + (µ p -1) 1µ ]

	and
	(ϕ 2 p ) ω = -	1 2N p Λ ϕ	(µ p -1)κ 2 i κ p ,	(170)
	(ϕ 2 + ) ω = -	1 2N Λ ϕ	κ s (κ 2 p + 4ω 2 ),	(171)
	(ϕ p ϕ + ) ω =	1 2N Λ ϕ	κ i κ 2 p (µ p -1) 1 -µ, (172)
	(ϕ 2 -) ω =	1 2N	κ s (κ i µ/4) 2 + ω 2 .	(173)
	where
	Λ ϕ 2	+
	+ω 2
	(169)

(ε 2 p ) ω = 2N p (µ p -1)κ i κ s κ p /Λ ε ,

(167)

(ε p ε + ) ω = -2N κ 2 s κ p /Λ ε ,

(168)

(ε 2 -) ω = -2N κ s κ 2 i (1 -3µ/4) 2 + ω 2 , 2 + +ω 2 (κ p + κ i µ/2) 2

(175)

7 Acknowledgements This work was performed within the French-Russian cooperation program "Lasers and Advanced Optical Information Technologies" with financial support from the following 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).


Evangelia Paxinou
email: evangelia.paxinou@gmail.com

The transparency of ambiances in architecture

Keywords: ambiances, transparency, design, back-plan, event

How ambiance is created in the frame of the architectural design process of public spaces? This paper shows that creating an ambiance is to design a dialectic relationship between the architectural elements that might constitute the back-plan and the ones that assure the potential emergence of the events, expressing thereby several levels of transparency as the transparency of senses and the transparency of practices. Based on the analysis of the research corpus of 25 projects of public spaces, mainly from Greece and France and also from other European countries such as Sweden and Spain, emerge several types of ambiance that cross projects, countries and cultures: the actual ambiance that exists due to the climatic, geographic, architectural and social conditions of the place and those that are not even there but they are waiting to be activated (virtual) by the individual perception and sensibility of users. These ambiances can become more identifiable and collective by the repetition of perceptual and perceptible phenomena. The paper presents the emerging attitudes concerning the creation of ambiance in public space. They are described by abstract nouns indicating the actions of managing the actual and emerging ambiances, such as the revelation, the reconfiguration and the merge. Through these actions are revealed the architectural intentions which will activate the affective dimensions of the project, such as the acclimatization, the appropriation and the challenging.

Transparency of senses

Every architectural project manages physical, social, sensible qualities of space and creates ambiance when experienced by future users. What kind of ambiance is created by the design? What do designers seek? If the ambiance is an overall and diffused feeling, then it needs a random event in order to awake the perception of its quality. So, the ambiance in the singular degree expresses a general and subtle feeling, while in the plural degree, ambiances are the expressions of the micro events that emerge from the dynamic interaction between the subject's sensibility and its context -emerging ambiances [START_REF] Paxinou | Αναδυόμενες ατμόσφαιρες στον αρχιτεκτονικό σχεδιασμό του δημόσιου χώρου[END_REF].

In other words, the ambiance can be considered as the transparent link between the designed space and the lived one. It filters architectural practices through the lived experience. It creates a general, stable, transparent back-plan which reflects the unexpected events. The events unfold in the front-plan and are activated by the interaction between users and the architectural design. This dialectical relation between the elements that assure the back-plan and the conditions that activate events determines the transparency between the architectural practices and the ones of the users [START_REF] Paxinou | Αναδυόμενες ατμόσφαιρες στον αρχιτεκτονικό σχεδιασμό του δημόσιου χώρου[END_REF].

Every place has an ambiance which arises from the combinations of its various environmental, architectural and cultural characteristics [START_REF] Norberg-Schulz | Genius Loci: Το πνεύμα του τόπου : Για μια φαινομενολογία της αρχιτεκτονικής[END_REF]. A person feels constantly in the back-plan of his body a general and indeterminate feeling [START_REF] Damasio | L'erreur de Descartes : La raison des émotions[END_REF]. Due to the transparency of the senses he cannot perceive immediately its quality. In order to understand what is happening he mentally processes the sensory environmental solicitations, according to his personal desires and experiences [START_REF] Merleau Ponty | Phénoménologie de La Perception[END_REF]. He selects and evaluates the senses that trigger his actions and awake his perception. This overall sensation is formed over time by the activation of unexpected momentary events. Due to their repetition, the events can become ordinary and they pass in the back-plan [START_REF] Thibaud | En Quête D'ambiances[END_REF] as the example of the human steps. During their activation, their intensity varies depending on the sensibility of the person who is experiencing them and they can pass on the front plan again. They can become remarkable and they provoke more intense emotions that will influence the overall sense of the place. So, the ambiance is the "medium" [START_REF] Böhme | Un paradigme pour une esthétique des ambiances : l'art de la scénographie[END_REF] which allows the simultaneous perceptions of the ordinary back-plan and the unforeseen events that emerge from it. In that sense, it enables the appearance of the architectural form -its perception by the users.

The paper shows that the handling of the simultaneous perceptions of the back-plan and the unfolding events is important for the architectural ambiances creation. It constitutes the activation mechanism of the design and enables the incorporation of the architectural intentions and the incarnation of the sensible experience. It encourages the innovation of the architectural design by the activations of events, either as unexpected -innovated architectural forms or by the improvisation of the users (unexpected behaviours). These reflections (simultaneous perceptions) express also the interaction between the existing diffused space and the ordinary interpersonal relations.

Therefore, «the conditions that assure the appearance of the ambiances» [START_REF] Böhme | Un paradigme pour une esthétique des ambiances : l'art de la scénographie[END_REF] are studied through the analysis of the conditions that constitute the backplan, the ones that assure the appearance of the events and their relations with their back-plan. Each category is analysed by the three interrelated levels of the lived experience: the immediate -sensory experience, the active sensible experience that makes sense and the affective experience -the activated emotions.

Regarding the conditions that assure the appearance of the back-plan on which unfold the sensible experiences (events), the following interrelated categories can be used [START_REF] Thibaud | En Quête D'ambiances[END_REF]:

-the «multi-sensory solicitations», -the «dominant tonality», -the «emotional tonalities».

During the design, the architect handles the material conditions which constitute the reflecting surface of the immaterial emerging complexities (ambiances). By the subject's sensible confrontation with the materiality of the space emerge «multisensory solicitations» which will open up the range of the users' perceptible potentials for actions, revealing or changing the dominant spatial perception. By the mental evaluation of the multi-sensory solicitations will emerge a «dominant tonality» colouring the existing ambiance. The dominant tonality predisposes «emotional tonalities» that affect the ways of being together in space [START_REF] Thibaud | En Quête D'ambiances[END_REF].

In the same way, the interrelated categories regarding the conditions assuring the appearance of the events are as follows:

-The sensory-motor materiality, which consists of the material conditions which activate the sensory -motor system of the subject (immediate bodily reactions).

-The sensible materiality is the material qualities which enable the appropriation of the event. They are triggered by the «expressive capacities of the body in-motion» [START_REF] Thibaud | En Quête D'ambiances[END_REF] and ensure the interaction between the architectural forms and the social behaviours.

-The affectivity expresses the bodily exteriorisations of the inner world of the person experiencing the event (behaviour changes, facial expressions, postures of the body), which influence the interpersonal relations and «colour» the ambiance of the space.

The above levels unfold according to their degree of objectivity and combine the objective with the subjective conditions of the ambiances. The first level gathers the objective material-physical conditions, the second refers more to the moral dimensions and the third describes the spontaneous behaviours and the unpredictability of feelings. Their analysis enables the comprehension of the dynamic relations between the back-plan and the events in the architectural design. What kind of relations are produced by the promotion of either the back-plan or the events? What happens when the two are involved and self-reinforced?

The public urban space is considered as the ideal research field for the study of the above dynamic relations, due to its unpredictability and the interactivity between its materiality and the unfolding individual and social activities. The ambiances emerge in the social space and influence its public character. The simultaneous perceptions of the ambiance/ambiances express the oscillation between the individual and collective experiences. The ambiances emerge as individual internal emotional expressions that become the expressive medium of the sense of place when they are exteriorised and ensure the social interaction.

Approach

The paper presents the results of a PhD thesis on the creation of an ambiance in urban public spaces. The research corpus consists of 25 projects of public spaces, located mainly in Greece and in France and also in other European countries, such as Sweden and Spain. These projects are mainly documented with semi-directive interviews with the creators.

The semi-directive interviews aim of the reactivation of the designing process around ambiances, by the perception of the experienced pre-linguistic phenomenon. The verbal formulation of the experienced senses provides the semiotic material (language and graphic) for the creation of the ambiances, which is subsequently translated into the architectural language.

The pre-selection of the creators is based on assumptions about their practices and their relations with the ambiances. How do they deal with the interactions between the material qualities and the action-perception of the users? The final selection of the projects was done in order to study a large panel of creation postures. The interviews were with Annie Tardivon (landscape architect, agency IN SITU, Lyon), Pascal Urbain and Thierry Cicionne (architects, agency STOA, Marseille), David Humbert (architect, agency Les Passagers des Villes-EAGD, Lyon), Jeppe A. Andersen (architect, Hellebaek), Prodromos Nikiforidis -Bernard Cuomo (architects, Thessaloniki), Sofia Vyzoviti (architect, School of Architecture, Volos), Philippe Liveneau (architect, School of Architecture, Grenoble) and Horacio Gonzales Dieguez (artist, noTours).

Even so, architectural practices of the selected creators can be roughly described as poetical, functionalist, political, sensitive, dynamic, technological, interactive, the analysis of the corpus, which belongs from different countries and cultures, shows that there are many common comments between the creators. Actually, they all express a form of a dialectic between the architectural elements that might constitute the back-plan and the ones that assure the potential emergence of the events. In addition, our analyse shows that this dialectic relation between back-plan and events can oscillate, whatever the project, between three states: the conflict, the negotiation and the sympathy. These categories are the same for the 25 projects studied. Therefore, their descriptions, concerning the influence of the competitive relations between the back-plan and the unfolding events, explain main differences between the public spaces studied whether in terms of their creation or their publicity [START_REF] Paxinou | Αναδυόμενες ατμόσφαιρες στον αρχιτεκτονικό σχεδιασμό του δημόσιου χώρου[END_REF]. Finally, the above relations express also the levels of transparency between the back-plan and the emerging events, a transparency that let elements emerge from the back-plan as unexpected events and the opposite.

Transparency of practices

The ambiances challenge the limits between the architectural practices and the users' activities because they have to be perceived by someone in order to exist. The design assures the conditions of the appearance of an ambiance [START_REF] Böhme | Un paradigme pour une esthétique des ambiances : l'art de la scénographie[END_REF]. The emerged complexities behind the transparent back-plan of an architectural object challenge the human senses and activate the potentialities of action (Gibson, 1977) which will affect the general ambiance. So the ambiances are not created only by the architect but also by the future users. This dynamic approach of the ambiances requires the transparency of the architectural practices in order to reveal the potentialities of the users' actions and encourage the improvisation, without disconnecting them from their spatial, sensible and social context. According to the above, the back-plan reflects the existing environment -the overall sense of the place where the architect is invited to create his ideas -events that solicit the users' senses and activate their feelings.

From the transversal analysis of the corpus of the 25 projects of public spaces, emerge different design postures that express and explain the main position of the creators in terms of their ambiance's creation. They are organized around abstract nouns indicating actions that handle the relation between the actual ambiance and the emerging ambiances (that will be created). The creation postures are the revelation, the reconfiguration and the merge. They describe the encounter of the ambiance on the back-plan with the ambiances on the front-plan. Either the design converges them into the same direction, revealing the existing general feeling, or it articulates them into new experiences which transform the existing, or finally it merges them and fuses the actual and the virtual.

Revelation of ambiances -acclimatisation

The revelation posture investigates the existing ambiance of space (back-plan) in order to reveal it and accentuate it by contemporary design methods (conditions of events' presence).

Back-plan:

It consists of the diagnostic of the place, in order to perceive its overall sense and of the creation of simple, ordinary architectural forms, in order to reveal and reinforce the existing landscape. Ephemeral tonalities emerge from the «negotiation» between the architectural intentions and the existing natural and historic landscape (sea, wind, light, vegetation, monuments, etc.). The design opens up the potentialities of actions and encourages the "synergy" of the multi-sensory solicitation [START_REF] Thibaud | En Quête D'ambiances[END_REF], in order to enable the re-appropriation of the space and the acclimatisation of the users.

Events:

The sensory-motor materiality of space is inspired by the physical space. The design encourages the improvisation by the creation of ordinary architectural forms soliciting microevents through their embodied experience (e.g the staircase that leads to the sea at Bo01 in Malmö) and the areas without predefined uses (e.g. linear lounge at Rauba-Capeù, Nice). Their materiality solicits various spontaneous behaviours and potentialities of actions. In this case, the events emerge from the social interaction which activate the materiality of the space. The potentialities of action are not completely predicted by the design, but they are predictable in practice (e.g. self-reinforced events). They emerge discreetly from the back-plan -as if they were always there1 -and activate feelings of appropriation and habituation (e.g. «it feels like home»). The creators are trying to «get into the mind» of the people, in order to understand how they feel2 . They work with the materiality that will excite them.

The ambiance is revealed for and from the public [START_REF] Thibaud | En Quête D'ambiances[END_REF]. The public embodies the back-plan through the spontaneous behaviours (events) which converge and create the collective overall sense of place.

Levels of transparency:

The design encourages the articulation of the perceptions of the existing back-plan and the emerging events. The existing ambiance is revealed through its reflections (events). Its revelations presuppose the exacerbation and the evolution of its characteristics. The overall sense is maintained, however, the project translates it into the contemporary construction language. Sometimes the events and the back-plan co-emerge (simultaneous perceptions) in order to acclimatize the users and make them feel as if nothing has ever changed.

Reconfiguration of ambiances -appropriation

The existing ambiance is transformed into a different one (which has never been experienced before). In our research corpus, it concerns mostly of places with no spatial quality. The design seeks for a new urban identity which will activate its public character.

Back-plan:

The organisation of the architectural and functional elements of the project aims to the architectural quality and the environmental balance and it forms a back-plan where the existing and disturbing ambient qualities are «corrected» (eg car noise, disturbing odours, poor lighting, etc.). In particular, by allowing nature to penetrate into the city emerge new urban ambiances. The architect subverts the existing activities and proposes new «scenarios» of activities3 . Most of the time the architectural intentions are focused on the design of a dominant sense, like lighting design, soundscape etc. Sometimes one sense is promoted (e.g. light or sound design) but this doesn't mean that the other senses do not participate in the perception of the spatial quality. Generally, the spatial quality is related to the emotions of belonging and habitation. In order to activate such emotions, the creator is inspired by the sense of the place and by his personal desires and fantasies and create a «scenography of feelings», based on «archetypes» of collective spaces (e.g. house -yard, living-room, moonlight, etc.).

Events: For the modulation of the ambiance, the architect is inspired by other architectural references and his personal experiences and he reconfigures the existing context into a new one (e.g. a meadow in the city). These reconfigurations often express his sensory relations to the characteristic phenomena of the landscape (light, water, sounds, odours, textures, etc.). He creates various sensible effects (lighting effects, sound effects) whose «reflections» on the back-plan destabilize the users4 . The architectural and personal references reflect the creator's artistic sensibility. His inner personal feelings are exteriorised and are converted to a common experience, in order to enable the subject's appropriation of the imaginary and the unfamiliar.

Levels of transparency:

The goal is the creation of sensible contrasts between the imaginary and the real world which express the personal and fantastic reflections on the architectural space. The contrasts emerge from the articulation of the back-plan and the unfolding events and they re-configure the overall sense of the place.

Merge of ambiances -challenging

The merge relates to projects where the existing ambiance is fused with the projected one, in a way that the one emerges from the other, or the one configures the other. Merging ambiances involves and creates an overall controversial feeling of a back-plan with fluid boundaries between what actually happens and what is about to happen. The design aims to represent the transformations of the architectural object within its dynamic field and the interaction between the actual and the virtual space.

Back-plan:

The architect creates parametric, unusual forms -«microarchitectures» which emerge from the dynamic interaction between the materiality and the corporality of the object and offer unexpected multi-sensory solicitations. The dynamic transformations of the multi-sensory solicitations encourage the creation of more eventful and ephemeral tonalities, which differ from the ordinary -everyday tonality of the public space.

In particular, the interactive technologies and the «augmentation» of the senses enable the merge of the different tonalities, activating a feeling of sympathy (Spuybroek, 2012). This feeling is developed by the corporation of the aesthetic relation between the architecture and the subject and determines the space quality, which means that someone feels nice when he becomes aware -he coordinates with the harmony of the space.

Events:

The creators are experimenting with the material and immaterial spatial qualities. They work with the aesthetic relations between the materiality of an object and its embodied experience (sensible materiality), in order to stimulate the senses and activate unusual and unpredictable emotions, depending on the subject's posture and movement -«ambient events»5 . These simultaneous emotions are created by microarchitectures that solicit microevents (spontaneous behaviours). The dialectical relation between the microarchitectures and the microevents enables the confrontation of the innovative and the ordinary forms, from which emerge the ambient events.

The completely unexpected events might emerge also in virtual environments, (e.g. like soundwalks). The interactive technologies create dynamically transformable effects which activate the merge of known and unknown experiences6 . When the effects are based on the active interpersonal relations (sensible materiality of the public) they activate the dynamic exchanges of experiences between people (e.g. artistic performance «Samba dress» by David da Paz and noTours). By augmenting the people's spontaneous behaviours, the users themselves become the events. They feel confused but not shocked. These feelings of confusion destabilise and trouble the participants and challenge them to discover the sense of place.

Levels of transparency:

The back-plan and the events co-emerge in a dynamic relation that varies according to time and create sensible contrasts between perception, action and existence. Through the experience of improvisation people are encouraged to choose when and how they will experience the space. In that sense real is what excites them (e.g. soundwalk in Gijon cemetery by NoTours).

Complementary actions

The above nouns describe general and structured postures and design approaches of the space-time and social characteristics of each project. They prescribe actions that encourage the emergence of the spontaneous, the instantaneous and the virtual elements. Because of their transparency, they can be applied separately or simultaneously at any project. In each case they activate the interaction between the back-plan and the emerging events.

Other complementary action nouns [START_REF] Torgue | Le Sonore[END_REF] are proposed in order to express the generalization methods of the nuances of the dynamic reflections between the back-plan and the emerging events. In another way, the following diagram combines the levels of transparency between back-plan and events with the basic design postures and complementary actions [START_REF] Paxinou | Αναδυόμενες ατμόσφαιρες στον αρχιτεκτονικό σχεδιασμό του δημόσιου χώρου[END_REF]. The events emerge in contrast, or within, or simultaneously with the back-plan. Each process is not hermetically defined, it oscillates with its adjacent in sensible balances, by flickering from the contrasts to the configuration and the coordination: the designer handles the material conditions activating events in order to reveal the existing back-plan, or to reconfigure it from ordinary to eventful and inversely, even to merge it with the instantaneous and the unpredicted.

For the revelation of the existing back-plan, he resorts to more direct actions combining place and time that will start to articulate, even merge, the back-plan with the events. Through the division and the densification of the events (dilution -gathering) he creates spaces with potential of actions. The revelations occur also by the control and the amplification of the active events, (recurrence), or their replacement, even their deletion (removal).

The reconfiguration of the back-plan requires more qualitative actions which will activate events based on the contrasts between the new and the existing (familiarunfamiliar), and on the articulations of the real with the imaginary. The contrasts are revealed through the isolation of events which are identified and distinguished. The articulations are achieved by the smooth passages of the events to the existing backplan (transition).

The action of decoration is oscillating between contrasts and co-emergences. The contrasts emerge when the decoration pursue the embellishment of the back plan (e.g. Disney Land) and the removal of the nuisances. The decoration which is inspired by the aesthetic relations between the back-plan and the events activate architectural events (e.g. micro-architectures). It expresses the medium for the creation of original events and unexpected emotions and it transits to the next approach, where both the existing back-plan and the events co-emerge. The assimilation of the back-plan by the events and vice versa (incorporation) engages also the corporality of the users (incarnation).

To conclude, the above table presents the synthesis of the analysis results of the research field (25 architectural and artistic works in public spaces) and can be considered as the theoretical tool (analytical grid) for the analysis of the ambiances of existing public spaces, and the design tool for the creation of the public space.

The dynamic variations of the relations back-plan/events reflect the emerging ambiances, as subjective sensible experiences depending on the sensibility and socio-cultural references of the users. They become collective and socials when they can be shared, through the repetition and duration of the perceptual and sensible phenomena.

Discussion

The emerging ambiances in architecture oscillate between two contexts: the sensible experience and the creation of space. The first includes the complexity of the in situ sensible experience that is, all the necessary dimensions for the immediate physical experience (sensory) and its understanding through the evaluation of the senses. The second context focuses on the creative act and the complexity of the sensible architectural and urban design, which places the individual in the environment and the society. The feelings produced by this sensible interaction between the human and its context depending on his perceptual ability, sensibility, and experiences, reflect the emerging ambiances of the space. The design of the emerging ambiances through the "game" between the back-plan and the unfolding events concerns all design practices and architectural postures, from the most «classics» up to the technological interactivity and unfolds the qualitative relation ambiance -ambiances. This dynamic relation reflects the dialectic relation between the creation and the experience of public space, through the approach of the time dimension (duration-instant) and the emotional dimension of architectural design (emotions -excitement).

This work does not impose a particular or effective design strategy which comes into conflict with the (contemporary) architectural trends and principles. It proposes the consideration of the ambiances through the handling of the dialectic relation back-up / events as an activating mechanism of the design process. Therefore, the main contribution of the ambiances in architectural design lies on their ability to affect, even to produce the project.

Particularly in public space, the notion of "public" imposes the study of different levels of transparency between the back-plan and the emerging events as their contrasts, their articulations and their co-emergence, in order to facilitate the dialogue between the dynamic and ethic performance of the ambiance phenomenon, proposing ways of existence and coexistence. It reveals, reconfigures, and merges the real with the virtual and the personal with the interpersonal.

The ambiance creation in architecture is a reachable horizon. The approach of architectural design through the interactive relation between the architectural elements that constitute the back-plan and those activating events provides the theoretical and practical tools for the creating of quality sensible experiences from which will emerge remarkable ambiances.

Figure 1 :

 1 Figure 1: (left) Ll'ambiance est là» at the Rauba Capeù waterfront in Nice and (right) Bo01 in Malmö. Sources: archives of 'STOA' and Jeppe A. Andersen.

Figure 2 :

 2 Figure 2: (left) The Feyzin Park in Lyon, by Les Passagers des Villes-EAGD and (right) The Garden of Seasons in Thessaloniki. Sources: archives of 'Les passagers des villes-EAGD' and 'P. Nikiforidis & B.Cuomo'.

Figure 3 :

 3 Figure 3: Couvent Sainte Cécil, expo. Peinture + Architecture, Grenoble (2010).Source: archives of Philippe Liveneau.

Table 1 :

 1 Emerging ambiances creation diagram

						Back-plan-events
	Contrasts		Articulation		Co-emergence		Contrast
		Reconfiguration		Revelation		Merge
	deco	isola	transi	conne	dilu	gathe	recur	remo	embo	incar	decora
	ration	tion	tion	ction	tion	ring	rence	val	diment	nation	tion

Pascal Urbain : «l'ambiance est là ».

Jeppe A. Andersen: «We found some way into people's mind and it suddenly gave us a way of thinking ».

David Humbert: «On se raconte des petites histoires et petit à petit on construit notre projet par la somme de ces petites histoires».

Jeppe A. Andersen: «The cool blue light transforms the public space and creates a different atmosphere of darker, starrier night skies».

Philippe Liveneau: «… le sujet et l'objet d'architecture émergent dans un même processus qui est l'évènement architectural».

noTours : «…they are not just interacting with the events, they have access with other events that you don't have access».