Nondegenerate four-wave mixing in a cavity: instabilities and quantum noise reduction

Volume 83, number 5,6

OPTICS COMMUNICATIONS

15 June 1991

Full length article

Nondegenerate four-wave mixing in a cavity: instabilities and quantum noise reduction M. Brambilla, F. Castelli, L.A. L u g i a t o , F. P r a t i a n d G. Strini Dipartimento di Fisica dell'Universit~ di Milano, via Celoria 16, 20133 Milan, Italy Received 23 January 1991

We consider the four-wave mixing interaction of three longitudinal cavity modes in a unidirectional ring cavity containing a system of two-level atoms. The set of equations which governs the dynamics of the system coincides with that of multimode optical bistability, and describes the behaviour of the system over all its parameter space. We identify an appropriate limit in which the model reduces exactly to a parametric interaction among the three modes. In this limit the model is fully quantum mechanical and it includes all the physical processes that are relevant in the limit (four-wave mixing, phase modulation, cross phase modulation). The stationary solutions of the parametric model are calculated analytically and display a complex scenario, which includes also the possibility of undamped oscillations of the modal amplitudes. Using the parametric model we calculate analytically also the spectrum of the fluctuations of the intensity difference between the signal fields, above the threshold of generation of the signals. It turns out that this coincides with the expression of the spectrum that is well known in the case of the nondegenerate parametric oscillator.

1. Introduction Four-wave mixing ( F W M ) is one o f the basic mechanisms in nonlinear optics. We consider the interaction o f a p u m p field o f frequency to1 and two signal fields (or more precisely, a signal and an idler field) o f frequencies o)2 and 0)3 such that to2+ tea = 2tol; hence the F W M process corresponds to the conversion o f two photons o f frequency to~ into two photons with frequencies 0)2 and tea, and vice versa. Especially interesting is the case o f a nonlinear material placed in a resonant optical cavity; the two signal fields, propagating along the longitudinal axis o f the cavity, can be spontaneously generated by the p u m p field. The mechanism is basically as follows: the p u m p field saturates the m e d i u m and, under appropriate conditions, any probe field with a frequency lying in a suitable interval experiences gain [ 1,2 ]. In a cavity, the modes work as probe fields, and one can obtain that a mode has a frequency that lies in the gain region; in addition, the p u m p gives rise to a F W M coupling o f the m o d e with another m o d e whose frequency lies on the other side with respect to the p u m p frequency [3,4]. As a result, the pair o f side modes experiences an overall gain and, when by increasing the p u m p intensity this gain becomes larger than the losses o f the modes, an initial r a n d o m fluctuation builds up the two side modes from zero, generating signal and idler fields which are nearly resonant with the frequencies o f the two cavity modes (in general, they are not exactly resonant because o f the frequency shifts generated by the refractive index o f the material). This process is an example o f a laser without inversion and has been experimentally realized few years ago [5,6 ]. The theoretical description o f the system, for values o f the p u m p intensity above the threshold for signal generation, is complicated because one must take into account the p u m p depletion, and therefore one must describe the attenuation o f the p u m p field during its propagation in the nonlinear medium. The problem bec o m e s simpler if the p u m p field is injected into an optical cavity, because in this case the dynamics o f the system is described in terms o f cavity modes. The p u m p cavity coincides with the signal cavity if the p u m p is collinear with the signals and it is different form the signal cavity otherwise. The simplest configuration is 0030-4018/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

367

Volume 83, number 5,6

OPTICS COMMUNICATIONS

15 June 1991

that of pump and signal fields which are collinear and copropagating in a unidirectional ring cavity (fig. 1 ), and quasi-resonant with three longitudinal cavity modes, precisely, the pump field is nearly resonant with a mode of frequency coc, the signal fields are close to resonance with the two side modes of frequencies o9¢_+or, where a is the free spectral range. This configuration of the system coincides with that typical of optical bistability (OB), and the phenomenon of side mode generation corresponds to the so-called multimode instability of OB. The total output field, i.e. the sum of the pump and signal fields, no longer has a stationary intensity when the side modes build up; the spontaneous intensity oscillations arise from the beating of the signal fields with the pump field. The multimode instability of OB was predicted several years ago [ 7 ], was then extensively analyzed theoretically [8-13 ] and was observed experimentally much more recently [ 14,15 ]. The multimode model was formulated in ref. [ 16 ], both in semiclassical terms and as a fully quantum statistical theory in terms of a multimode master equation; the Fokker-Planck approximation of the master equation was worked out in ref. [ 17 ]. The phenomenon of FWM is also one of the most important mechanisms to produce nonclassical states of the electromagnetic field [ 18-28 ] and, as a matter of fact, the first experimental observation of squeezing was achieved in a FWM configuration [ 29]. Other experiments were performed later [ 30,31 ]; a recent paper [ 32 ] reports on the observation of quantum noise reduction in the intensity difference between the signal and idler fields, similar to the phenomenon first studied and experimentally observed in a nondegenerate parametric oscillator [ 33,34 ]. Nondegenerate FWM in a cavity can be described by various models with different levels of sophistication and of completeness. Attractive for its simplicity is the model formulated by Savage and Walls [23 ], which describes the phenomenon as a parametric process with an interaction hamiltonian of the form

nFw~ =hg( a2 a~a[ + a~2 a2a3 ) ,

(1.1)

where al, a2, a3 (aT, a ~ , a~) are the annihilation (creation) operators of photons of the three cavity modes associated with the pump field (a~) and with the two signal fields (a2, a3), respectively. This description is similar to that which is commonly used for optical parametric oscillators. However, in the case of FWM the nonlinear materials is not a crystal but an atomic medium. Hence other processes, such as, for example, nonlinear absorption and phase modulation, play a role in the system behaviour and they are not contained in the model [ 23 ]. In order to describe the system adequately one must include the atomic variables in the picture. Models which fulfd this requirement have been given in refs. [21 ] and [22] and have been used for comparison with the experiment of ref. [ 29 ]; they neglect, however, pump depletion and therefore hold only belov, the threshold for signal field generation. These models coincide with the approximation of the multimode OE model [ 16], obtained by treating the pump field as a given constant (i.e. neglecting the pump depletion) and the signal fields to lowest order. The model [21 ] has been generalized [24] to include also inhomogeneou,, broadening and the case of the pump being noneollinear with the signals. A model which is valid also above threshold has been formulated recently [26 ], and includes also the possibility of inhomogeneous broadenin~ and noneollinear pump. In the case of homogeneous broadening and collinear pump, the model of ref. [26] coincides with the approximation of the multimode OB model of ref. [ 16 ], obtained by the following steps (i) one considers only three modes for the electric field, the atomic polarization and the atomic populatior Er

/ /

~

0

V

E2 ~\ \ \---4E~

k

2

4N

A

/

368

Es

Fig. 1. Ring cavity with plane mirrors. Mirrors 3 and 4 are as sumed to be ideal reflectors, while mirrors 1 and 2 have powc transmission coefficient T. E! is the incident field and it has stationary intensity and phase; the output field is composed, il general, by a field E~ with the same frequency t~t of the inpu field, and the two signal fields with frequency t~2 and 0)3 sucl that 0)2 +023 =2o)i.

Volume 83, number5,6

OPTICS COMMUNICATIONS

15 June 1991

difference, and (ii) in the final dynamical equations, one neglects all terms which are of order higher than second in the signal modes. Because of approximation 2 in the above-threshold case the theory [26] can be used only in the neighbourhood of the threshold. The analysis of this paper proceeds along two parallel lines. On the one hand, we want to use the model of ref. [ 16 ] to describe the quantum fluctuations and the instabilities above threshold, in principle over the whole parameter space of the system. On the other hand, we determine under which conditions the complex model of ref. [ 16 ] reduces exactly to a model which describes the phenomenon in terms of parametric processes involving the three modes, thus providing a picture with a simplicity comparable to that of the model of ref. [23 ]. The model we derive, however, is not assumed ab initio but is derived exactly from the theory [ 16 ] by introducing appropriate limits and, furthermore, it includes all the physical processes that are relevant in this limit. We check the validity of our parametric model by performing a direct comparison between its solutions and the numerical solutions of the complete model of ref. [ 16 ]. Our analysis concerns both semiclassical and quantum statistical features. In the semiclassical theory we study the steady states, the instabilities and the dynamical aspects. We consider first the instability which gives rise to the build-up of the signal fields, and calculate analytically the multimode stationary solutions that emerge from this instability when the system is driven above threshold. Then we analyse the instabilities which arise when the input intensity is increased well beyond the threshold value. In the quantum statistical treatment, in this paper we calculate only the spectrum of the intensity difference between the signal fields, which is the quantity observed in the experiment of ref. [ 32 ]. The other quantities of interest (e.g. the squeezing spectra) can be obtained straightforwardly from the parametric model. For the sake of simplicity, we restrict our treatment to the case of homogeneous broadening and collinear pump. The generalization of our parametric model to the cases of inhomogeneous broadening and noncollinear pump can be obtained easily by following the procedure indicated in ref. [24 ] and [26]. In section 2 we review the multimode model for OB, the single-mode stationary solution and its stability analysis. Section 3 defines the parametric limit and derives the parametric model. In section 4 we calculate analytically the semiclassical multimode stationary solutions of the parametric model and study their instabilities. We compare these results with those of the complete model [ 16 ] and of the Savage-Walls model [ 23 ]. In section 5 we calculate analytically from the parametric model the spectrum of the fluctuations of the intensity difference between the signal and idler fields and, again, we compare it with that predicted by the theories of ref. [ 16] and of refs. [23,27]. The final section 6 includes a discussion of the results and the conclusions.

2. The muitimode model

We consider (fig. 1 ) a coherent, linearly polarized electric field EI of frequency to~, which is injected into a ring cavity of total length ~e, containing a homogeneously broadened system of two-level atoms of length L. We call toa the atomic transition frequency, 7J. the relaxation rate of the atomic polarization and yj the relaxation rate of the atomic population difference. The configuration of the system is described by the electric field ~f, the atomic polarization ~ and the difference D between the population of the lower and the upper level, normalized to the number of atoms. In the plane wave approximation, one introduces the envelopes F and P defined by the relations (k~ =to~/c)

8(z, t) - hv/~xTu ½[F(z, t) g

ei(/czz-ant)+c.c.]

¢(z, t) = ~l ~N -P [e(z, t) ei(k'z-~"~+c.c.],

,

(2.1a) (2.1b)

where ~ denotes the modulus of the dipole moment of the atomic transition and p is the atomic density. The 369

Volume 83, number 5,6

OPTICS COMMUNICATIONS

15 June 1991

coefficients appearing in eqs. (2.1) are introduced in order ot minimize the number of parameters which appear in the final equations. The dynamics of the system can be formulated in terms of the modal amplitudes by introducing the following transformation of independent variables [ 16,12,35 ]: z'=z,

L#-Lz t'=t+ - - c L'

(2.2)

which converts the atomic sample into a ring. In this way, the fields F, P, D can be expressed as follows [ 12,35 ]: F(z', t' ) = e x p { - ( z ' / L ) [In(1 - T)-i~ol}

( - T Y + "2

/

e~2~z'/L e-i~"fn( t' ) ,

(2.3a)

-t-oo

P(z',t')=exp(-(z'/L)[ln(1-T)-i~o]}

~, nm

ei2~z'/Le-i~"rp~(t'),

(2.3b)

~oo

+oo

O(2', t') =

Z nm

ei2nnz'/Le-i""Cdn( t' ) ,

(2.3c)

--co

where T is the power transmissivity coefficient of the mirrors (see fig. 1 ); 5o is the quantity 5o = ( Og¢--Ogl ) l ( cl Le) ,

(2.4)

with 09¢ being the frequency of the cavity mode lying closest to the input field frequency og~.The parameter Y denotes the normalized input field, defined by El(t) = hx/~l rMT ½y(e_iO~,t+ c.c. ) , #

(2.5)

where, for the sake of definiteness, we assumed that Y is real and positive. The frequency of the mode labelled by n (n=0, + 1, +2, ...) is given by to¢+txn, where an=not,

ot=2nc/L#,

(2.6)

and ot denotes the free spectral range. Because the population difference D is real, one has that d _ n ( t ' ) = d*~(t'). The equations for the modal amplitudesf~, pn, d~ are complicated in the general case; they become simplel if one assumes, as usually, the uniform field limit [ 16,12,35 ] defined by the conditions aabL<
T<
t$o<
(2.7a)

with C-OtabL/2T

arbitrary,

O=_Jo/T=(to¢-tol)/x

arbitrary,

(2.7b)

where Otabdenotes the linear absorption coefficient of the electric field and r is the cavity damping constant x = c T / Z? .

(2.8)

C and 0 are usually called the bistability parameter and the cavity detuning parameter, respectively. The time evolution of the modal amplitudes in the limit (2.7) is governed by a master equation formulated in ref. [ 16 ] (see also ref. [ 12 ] ). When the number of atoms is large, the master equation can be approximated by a Fokker-Planck equation for the quasi-probability distribution of the system in the generalized P-repre. sentation [ 36,37 ]. The Fokker-Planck equation is derived in ref. [ 17 ]. Strictly speaking, because the diffusior matrix is not positive definite, the Fokker-Planck equation should be formulated in the positive P-represen. tation [ 38 ]. However, in this work we will calculate only semiclassieal stationary solutions; all these quantitie,, 370

Volume83, number 5,6

OPTICS COMMUNICATIONS

15 June 1991

do not require the use of the positive P-representation. The Fokker-Planck equation in the positive P-representation is equivalent to a set of Langevin equations for the modal amplitudes, which take the form [ 12,16,17 ] df./dt' = - r [ ( 1 + i0)f. - Y6.,o + 2Cp.],

(2.9a)

df*./dt' = - x [ ( 1 - i O ) f * - YS.,o + 2Cp*],

(2.9b)

dp./dt' =Yl ( ~ f.. d . _ . , - [ 1 + i (A-o~./y± )p.] ) + F~.(t'),

(2.9c)

dp*/dt'=y± ( ~ f . ,* d . , . - [ l - i ( d - o ~ . / y ± ) p * ]

(2.9d)

) +Fu.(t') ,

dd./dt' = -Yll ( ½ ~ ( f "_.,p._., + f ~ , p .*, _ . ) + d . ( 1 - i a . / ? ~ ) -6..o) +F~.(t' ) ,

(2.9e)

n'

where A is the atomic detuning parameter '4= (Wa --Og~)/y± ;

(2.10)

in the following we will always assume for definiteness .4 >/0. The quantities F represent quantum noise terms and describe stationary, gaussian stochastic processes with zero average and haying time correlation functions , , = (F~.(t,)Fp.,(t2)) t

,

','±Yu ~f.op.+.,_.oO(t'l-t'2) 4C~Ns . Y.I. Yll

~'~

.

*

t

(2.1 la) ,

( Fp.( tl )Fp~( tz) ) = - 4CxNs ~',.f .. p.+.,_.,,6( t l - t 2 ) ,

(2.1 lb)

(Fp~(t'~)F...(t'2)) = yj. (2y± -y,,) ( ~ . . . , - d . , _ . ) 6 ( t ] 7 t [ ) 2Cr.Ns

(2.1 lc)

' Y~ ( 6 n + . , , o - - d . + . , + ½ 2.,,( f * . . p . + n , _ . . + f ~ . p * , , _ n _ . , ) ) 6 ( t ' l - t ' z ) (Fa.(t,)Fa., (t'2) ) = 4CxNs

(2.1 ld)

( Fp.( t', )Fd., ( t'z) ) = ( Fp.( t'l )Fd.. ( t'z) ) = 0 ,

(2.1 le)

where Ns is the saturation photon number. The semiclassical equations are obtained by dropping the fluctuation terms F. They allow one single-mode stationary solution given by [39,40] f s =fos6..o , P.s =PoS.,o s , d.s = dos 6.,o

(2.12a)

with (I-iA)fos pos= 1+.42+ l/Sl 2 ,

1+.4 z dos - 1+.42+ ifosi2,

(2.12b)

and f os obeys the stationary equation 2C .fos iz) + i ( 0 Y---fos [(1 + 1 +'42+ ]] 1 +.422C.4 + If s 12,//•

(2.12c)

The linear stability analysis of the single-mode stationary solution is obtained by introducing the deviations with respect to the stationary value, 371

Volume83, number 5,6

8 f ~ ( t ' ) = f ~ ( t ' ) - f s,

OPTICS COMMUNICATIONS

8 p . ( t ' ) = p . ( t ' ) - p s,

8 d . ( t ' ) = d ~ ( t ' ) - d s,

15 June 1991 (2.13)

and by linearizing the semiclassical equations with respect to the deviations. The infinite set of linearized equations splits into separate blocks of five equations for the variables 8f., 8 f * . , 8p., 8p*., 8d.. Assuming the socalled good cavity limit, x<< Y_~,Y,,

(2.14)

which is typical of multimode instabilities, the atomic variables 8p., 8p*., 8d. can be eliminated adiabatically and we obtain a closed set of equations for the field deviations 8f., ~ f * . , which are [41 ] ( d / d r ) S f . = - x { [ 1 +iO+2CT, ( a . ) ]Sf. + 2CT2 ( a . ) 8f*__.},

(2.15a)

( d / d t ' ) S f * n = - - X { [ 1 - i O + 2 C T T ( - a n ) ] 8f*__, + 2 C T ~ ( - a , ) 8f~},

(2.15b)

where the functions T1, T2 are defined by ( 1 -I-d 2 ) ( 1 -iA-iot,,/y. ) + i ( a . / 2 y . ) D'l If s 12/(711 - i a . ) ] ( 1 +iA) Tl = -- ( 1 +A2+ If s [2)(1 - i o t . / 7 . )2+d2+ [Yll/(Yll- i a . ) ] (1 +d2+ [fs [2)( 1 -iota~y±) ' T2=l

y,,(fs)2 ( 2 _ i a , ) 1--iA ii_iot, ~- l+d2+lfSl2.

(2.16a)

(2.16b)

Eqs. (2.15a), (2.15b ) coincide with the semiclassical version of the FWM model formulated in refs. [ 21 ] and [22 ]. The quantities 2CT~ and 2CT2 coincide with the absorption/gain coefficient and with the four-wave mixing coupling coefficient, respectively, introduced in refs. [ 3 ] and [ 4 ]. The coefficient T~ (or,) governs the gain experienced by the mode n alone, while the coefficient T2(a,) generates the coupling between modes n and - n which alters the gain or absorption experienced by each mode. The ansatz

(8s.(,,)

(v<.o, 3

8f*.(t')]

\ S f *(°) ]

(2.17)

introduced into eqs. (2.15) leads to a quadratic equation for 2, whose roots are [41 ] 2~") = - x [ 1 +2CT'~ (or.) +_~rl/Z(ot.) ],

(2.18)

where we have introduced the symbols

T'~(a.)=(1/2)[T~(ot.)+TT(-ot.)],

T'{(ot.)=(1/2i)[T~(a.)-TT(-a.)],

(2.19)

~ ( ot.) =4C2T2( a.) T~ ( -or.) - [ 0 - 2CT~ (or.) ]2. The multimode instability arises when one of the two eigenvalues (2.18) has a positive real part for at leasl one value of the modal index n ~ 0. Hence, if we set if+ (otn) = R e [ -T-~l/2(O~n) -2CT'l (or,,) ] ,

(2.20)

the instability condition takes the form ~+(a.)>l

or

~_(a.)>l,

(2.21)

for at least one value of n # 0. The functions (¢+ and (~_ represent the overall gain/absorption experienced b3 the pair of side modes (n, - n ) in a stationary state in which only mode n = 0 is active. Eqs. (2,21) impos~ that the overall gain is larger than the losses experienced by the modes. In order to obtain the phenomenon of the spontaneous growth of a pair of side modes (n, - n ) , it is necessa~ that one of eqs. (2.21) is satisfied for a value n # 0 and simultaneously that eqs. (2.21) are not satisfied fol 372

Volume 83, number 5,6

OPTICS COMMUNICATIONS

15 June 1991

n=0. Clearly, this is impossible in the limit or=0 (i.e. a , = 0 ) which, in the FWM literature, is called the degenerate case. Note that this is an exact result which follows from the complete model (2.9). The same result remains true for the parametric model that we will derive, whereas it is not valid for the model derived in ref. [261. For the description of the system above the threshold for generation of the signal modes, we will limit our considerations to the three field modes fo, f+l, f-~ and accordingly in eqs. (2.9a, b) we will consider only the case n = 0 , _+ 1 and in eqs. (2.9c, d, e) and (2.11 ) the sums will be restricted to n', n" =0, + 1. This model, where we have only three field modal variables, fo, f+~, f-~ and an infinite set of atomic modal variables p,, p*, d, (n = 0, _+ 1, + 2, ... ) will be called in the following model A. We will consider also another model (called model B), in which one includes only the modal variables with n = 0, - 1, + 1 also for the atomic quantities. If we use the indices 1, 2, 3 instead of 0, - 1, + 1 respectively, as is commonly done in the FWM literature, the dynamical equations for model B take the following explicit form:

d f / d t ' = - x [ ( l +i0)ft - Y+2CPl ] ,

(2.22a)

df2/dt' = -~c[( 1 +i0)f2 + 2 C p 2 ] ,

(2.22b)

dfa/dt' = -to[ ( 1 + i0)f3 + 2Cp3] ,

(2.22c)

dpt/dt' =y± [fl dl +Ad2 +f2d~ - (1 +iA)pl ] + Fp, (t' ) ,

(2.22d)

dPE/dt' =y± Lf~d2 +fzdl - ( 1 + i J + i a ) p 2 ] + F ~ ( t ' ) ,

(2.22e)

dp3/dt' =Yx [J] d'~ + f 3 d l

( 1 +id--it~)p3 ] +Fp3(t'),

(2.22f)

dd,/dt'= -~,g~[½(f~; Pt + f'~P: + f'~ P3 +P~;Z +P'~A +P~A) +d~ - 1 ] + F,~,( t ' ) ,

(2.22g)

dd2/dt' = -~'u[ ~I (f~• P2 +f3P~ • +P~A +P~A) +d2 ( 1 +iot/yll) ] +Fd:(t' ) ,

(2.22h)

--

the equations for f 7, f ~, f 3, PT, P~, P~ and d3 correspond to the complex conjugates of eqs. (2.22a-f) and (2.22h), respectively. The quantity t~ is defined as c~=ct/y±.

(2.23)

The nonvanishing correlation functions are

(Fm(t'l)Fpl(t'2))=(Fp3(tl)Fp2(t'2)}=

- Y-- Yll @Pl +f3P2 +fzP3) 8(t'~ - t [ ) , 4CtcNs

17 ' ' (p,(t~)Fp,(t2)) =

Y±Yll 4CxNsfP, O(t'~-t'2)

(I'p, (t'~) Fp~(t'2) ) =

YJ-;'u ~p~+fip,),fft'~-t'2) 4CKNs

(2.24b)

(i=2, 3),

, Fp~(t,2) ) - ~ ' l 2CtcNs (2y.-~q) (l-d,)a(t;-t'2) (Fp,;(t~)

-~q) diS(t'~ -t'2) (Fpr(t'~) Fp,(t'2) ) = - y±(2yj_ 2CxNs

(2.24a)

(i=2,3),

(2.24c)

(i=1,2,3)

(2.24d)

(i=2, 3 ) ,

(2.24e)

(I'd, (t'~) I'd, (t'2) ) = 4CXNs [ 1 -d~ + ½( f TPl +f'~P2 "+f'~P3 + C . C . ) ] d~(t] - - t [ ) ,

(2.24f)

(Fa2(tl)Fd2(t~))-- 8 ~Y~ (f2P~+f~P2) 8(t]

(2.24g)

--tl),

373

~rolume 83, number 5,6

OPTICS COMMUNICATIONS

Y~ [ - d 2 + ~' (f2P,* + f *3P, +fl P3* + f T P z ) ] 6(t'~ -t'2) " (Fa, (t'l) Fd2(t'2) ) - 4CxNs

15 June 199 ! (2.24h)

the correlation functions (FvT(t'~) FvT(t'2 ) ) etc. are obtained by taking the complex conjugate of eqs. (2.24a) etc. The analysis of refs. [21,22,24,26] is based on model B.

3. The parameter limit and the parametric model In order to obtain a parametric model, it is necessary to perform the so-called cubic approximation, so that the final dynamical equations for the modal amplitudes, derived by performing the adiabatic elimination of the atomic variables under the condition (2.14), display only cubic nonlinearities. According to the singlemode theory of OB, the conditions for the cubic approximation are as follows [42,43 ]: ( 1 ) The atomic detuning parameter A must be large, in the sense that d>>l,

Ifl2/A2<
(3.1)

(2) In order to ensure a meaningful nonlinearity in spite of the large detuning, the bistability parameter C must be large, in such a way that

2C/A>> 1 with ( 2 C / A ) I f IZ/d 2 arbitrary.

(3.2)

(3) The cavity detuning parameter 0 must be large, so that 0 >> 1 with 01 -- 2 C / A - 0 arbitrary.

( 3.3 )

(4) Furthermore in order to make the absorptive effects negligible, we require that

2C/A 2 << 1 .

(3.4)

We observe that the situation of large values for the parameters C and A, satisfying conditions (3.1) to (3.4) is met, for example, in the experiment of ref. [ 32 ]. In the multimode case, we must ask that the same conditions hold also when A is replaced by the detuning parameters for the side modes AT- ~. This requires the further condition

~t/A<< 1 ,

(3.5)

which ensures also that the nonlinear terms of the final dynamical equations do not depend on a. We note, however, that one cannot simply take ot = 0 because, as we discussed in section 2, in this case there is no multimode instability. All the previous conditions can be summarized and made precise by introducing a smallness parameter ¢ (a posteriori, one can verify that the final parametric model remains valid also when e is not very small, for example, E= 1/4). We assume that

~t/A~,

2 C / A ~ e -2 , Ol=_2C/A-O~e °,

A~e - s ,

(3.6)

and look for solutions such that ]f [2/A 2 ~ e 2, which implies that (2 C/A) ( If[ 2/A 2) ~ e o; these conditions make the cubic approximation rigorous. By combining the first and the second of eqs. (3.6), one obtains thal (2C/A) (rt 2/A 2) ~ ¢o; this condition implies that the parameter oe appears in the final equations up to second order. The presence of the second-order contribution in t~ is necessary to ensure the possibility of a multimode instability. Note that the following treatment does not change if the fourth of conditions (3.6) is replaced by A ~ e -p with fl> 3. Let us consider first the model B, given by eqs. (2.22). We perform the adiabatic elimination of the atomic variables by dropping the derivatives with respect to time in eqs. (2.22d-h). Then we calculate the polarization 374

Volume 83, number 5,6

OPTICS COMMUNICATIONS

15 June 1991

p, (i= 1, 2, 3) to zeroth, first and third order in the field amplitudes f. The zeroth-order contribution arises from quantum noise, 1

p~O)=iy±dFp ' ( i = 1 , 2 , 3 ) ;

(3.7)

in accord with the first and the fourth of conditions (3.6), eq. (3.7) is obtained by neglecting ~ With respect to d and 1 with respect to ia. It turns out that (3.7) is the only noise contribution that is relevant in the limit (3.6). The first-order contribution is

p~.) =f,/ia, p~,)= ~/ia)(1-~/a+a2/a2),

p~'~= ~/ia)(l

+e,lA+~Va2),

(3.8)

where, again, we kept only the contributions which lead to terms, in the final equations for the amplitudes f, which are nonnegligible in the limit (3.6). Finally the third-order contributions are 2+i~ )] i:.cc2+(a2/y±y,) l + 2 ( l + i o t / y , ) +~-~Ju2:3 ~ ~ ,

(3.9a)

+-~f3f~2(l+ia/yN), i. 2+it~

(3.9b)

i r * ' 2 2(1-iot/y,,) 2-it~ + If212+ If312 ] + "~a2Jl

(3.9c)

iAf_3~[ ( 2-it~ ~ If, 12+lf2l 2 l + 2 ( i - - i a / y , ) / - I f 3 1 2

(

pp)=a,klf,iA[ 12( 1+ 2(l+iot/y,,)] 2+it~ ~+lf212+lf3l

2]

p~3)=

pp)-- a3klAif3F12 ( 1+ 2(l-iot/y,).l 2-it~ ~

Next, we insert eqs. (3.7), (3.8) and (3.9) into eqs. (2.22a, b, c). In the following, we will consider only the purely radiative case 7, = 2 y ± , which is the most interesting from the viewpoint of quantum noise reduction; in the case 711<< 7± there is no multimode instability in the limit (3.6). If we introduce the change of variables Z =A,

f2=A

exp[+i(2Cx~/j2)t], A=f3 exp[-i(2Cx~/A2)t],

(3.10)

the final equations take the form

df~/dt' =x{Y-f~ [I -i01 +i(2C/zP) ( If~ 12+2 If212+2 If312) ] -i(4C/A3)fTfJ3}+F, (t'), df2/dt' = x{-f2 [ 1 -i02 +i(2C/A 3) (2 If~ 12+ If2 [2+ ~/If3 12)1 -i(2C/A3)f~f2}+F2(t ' ), df3/dt' = x{-f3 [ 1 -i02 +i(2C/A 3) (21f~ 12+ r/lf215+ I f 3 1 2 ) ] - i ( 2 C / a 3 ) f ~ f ] } + F 3 ( t ' ) ,

(3. l la) (3.1 lb) (3.1 lc)

where we defined 02------Ol+

(2C/A)(a2/gl 2) ,

(3.12)

and r/is equal to 1 (the reason for introducing the parameter ~/will be clear in the following of this section). The quantum noise terms F,- are given by

Vi=i(2Cx/7±a)V,,, (i=1,2, 3) ;

(3.13)

their time correlation functions are obtained from eqs. (2.24a-e), in which one replaces p; by (f/iA) [see the first terms in eqs. (3.8)]. One obtains (/'l (t])F~ (t[)) = (/'2 (t])/'3 (/[)) = - i g 0 r2 + 2f2f3)6(t'~ - t ~ ) ,

(3.14a)

(I'~(t'~)I'~(t'2)) =-igf26(t'~ -t'2) (i=2, 3), (I'~(t'l)I'~(t'2) ) =-2i~fJ(t'l -t'2) (i=2, 3), (I'*(t'~)I'j(t'2))=O ( i , j = l , 2, 3 ) ,

(3.14b) (3.14c) (3.14d) 375

Volume 83, number 5,6

OPTICS C O M M U N I C A T I O N S

15 June 1991

where we set g=2Cx/A3Ns .

(3.15)

The time evolution equations for the variables j r . ( i = 1, 2, 3 ) are obtained from eqs. (3.11 ) by complex conjugations; in the same way one obtains the remaining time correlation functions from eqs. (3.14a-c ), The parametric equations (3. l 1 ) with (3.14) are an exact consequence of the initial model B in the limit defined by eqs. (3.6). If one starts from model A instead of model B and follows the same procedure, one arrives exactly at the same parametric model (3.11 ), (3.14) with the only difference that in this case r/= 2. This difference arises from the inversion variables d2, d_ a (using the nomenclature for which the three modal amplitudes are fo, f_+~), which are neglected in model B. Let us now consider the density operator p of the three-mode system and the master equation 3 i dp _ ~,, A i p - -~ [H,p] , dt' i=1

(3.16)

where the liouvillians Ai are given by Ai = x ( [ai, pa*, ] +[aip, a*, ] ),

(3.17)

and the hamiltonian H is defined as H=H~xt + Ho + Hi,t ,

(3.18)

with Hext =ihxoq(a~ - a l )

,

(3.19)

Ho = -hX[Ol a]al +02(a~a2 +a~a3) ] ,

(3.20)

Hint =HvwM +Hp~a +HcPM •

(3.21)

The hamiltonian HvwM is defined by eq. (1.1), while HeM and HcpM are given by

//PM= ~g

a*,2a2

(3.22)

i=l

Hcr, M =2hg(a~al a~a2 +atala~a3 +a~a2a~a3) ,

(3.23)

with the coupling constant g defined by eq. (3.15). The master equation can be converted into a classical-looking Fokker-Planck equation by using the P-representation; the equivalent set of Langevin equations coincides with eqs. (3.11 ) with q= 2 (i.e., it coincide,, with the parametric model derived from model A); the correlation functions coincide with those given in eqs (3.14). This fact can be easily verified by taking into account the relations fii=oti/x~s,

y=oq/x/~s ,

(3.241

where ot~ ( i = 1, 2, 3) denote the c-number variables associated with the operators a~. The hamiltonians Hp~ and HcpM describe self phase modulation and cross phase modulation, respectively, as one normally has in X~3: media. The cross phase modulation terms have the same form of those which arise in the analysis of three level media [ 44 ]. The simplest parametric model of ref. [23 ] amounts to eq. (3.21) without the phase modulation contri butions HpM and HcpM. In our theory eq. (3.15) provides the microscopic expression of the coupling constant The same is true for the parameter 02 defined in eq. (3.12); note that, because ot must be necessarily large 376

Volume 83, number5,6

OPTICS COMMUNICATIONS

15 June 1991

than zero in order to have signal generation, the detuning parameters 01 and 02 must be different.

4. Stationary solutions of the parametric model linear stability analysis and dynamical solutions. Comparison with the solutions of model B The form of eqs. (3.11 ) suggests immediately the introduction of the normalized variables

xi=x/~a~=(2C/A3)l/2f~

(i=1,2, 3),

Y=x/g/xa~=(2C/A3)l/2Y,

(4.1)

so that eqs. (3.1 1 ) become

dxl/dt' =x{ ]~-xl [ 1 -i01 + i ( Ixt 12÷21x212÷21x3

12) ] --2ixTx2X3}+l"x, ( / ' ) ,

(4.2a)

dx2/dr = x { - x 2 [ 1 -i02 +i(21x1 12+ Ix2 12+~/Ix312)] -ix~x2}+F~(t'),

(4.2b)

dxa/dt' = x{-x3 [ 1 -i02 + i ( 2 Ix1 I2+~lx2 let Ix3 I2) ] --ix~x~)+rx~(t'),

(4.2c)

with

Fx, = (2C/A 3) 1/21~i (i= 1, 2, 3 ) . Correspondingly, the time correlation functions of the stochastic processes Fx, (i = 1, 2, 3) are identical to eqs. (3.14a-c), with the quantities f/replaced by xi (i = 1, 2, 3 ). As one sees from eqs. (4.2), our parametric model depends on three effective parameters; the normalized input field I? and the detuning parameters 01 and 02. In the remaining part of this section we consider eqs. (4.2) in the semiclassical approximation, i.e., we drop the quantum noise terms Fx, (i= 1, 2, 3). The semiclassical equations admit one single-mode stationary solution given by the well-known cubic steady-state equation [45 ] ]~=xS[ 1 -i(01 - IxSl2) ] ,

(4.3)

which predicts bistability for 01 > v/3. The signal fields x2 and x3 vanish in this solution, which corresponds to the case below threshold for signal generation. Figs. 2a, 3a and 4a show the comparison between the exact single-mode steady-state curve, given by eq. (2.12c), and the cubic approximation (4.3). In the case of the model of ref. [23] one has simply xl = Y. Let us now perform the linear stability analysis of eqs. (4.2) around the single-mode stationary solution, by introducing the deviations ~x~ =x~ - x s ,

(4.4)

where x S = 0 for i=2, 3, and x s is a solution ofeq. (4.3). The linearized eqs. (4.2) are (d/dr)~x I

~[ ( 1 --i01 ÷ 2i Ix s 12) ~xl + i ( x S ) 2 8 X T ] ,

(4.5a)

(d/dt')SxT = - x [ ( 1 +i01 - 2 i Ix s 12) ~ixT-i(xS*)2~xl ] ,

(4.5b)

(d/dr)8x2 = - x [ (1 -i02 + 2 i l x s 12) 8x2 +i(xS)28x~],

(4.5c)

~-~

--

( d / d r ) S x ~ = - x [ (1 +i02 - 2 i l x s 12)~x~-i(x~)28x2].

, ,

(4.5d)

Note that eqs. (4.5) do not depend on r/. The two variables 8Xl, 8xT obey the self-contained set ofeqs. (4.5a) and (4.5b); the same is true for the pair of variables 8x2, 8x~, which obey eqs. (4.5c) and (4.5d); the two variables 8x3, 8x~ obey the complex conjugates of eqs. (4.5c) and (4.5d). By introducing the ansatz

8x*]

\Sx.tO) I (i=1, 2, 3) , 377

Volume 83, number 5,6 I

OPTICS COMMUNICATIONS I

I

I

I

I

I

I

I

I

I

I

I

I

I

b

3.

I

I

15 June 1991

i/llleFIi/t/plllIJll~

3.

&

I xSl 1

2. 2.

1. I.

0.0

I

O. 0

I

,

I

L

I

I

10.

t

I

t

I

t

20.

0.0

I

30.

,,, 0.0

9

,

I

1.

,

t

I

I

I

2.

I

t

t

I

I

I

I

I

S.

4.

ixSt I

Fig. 2. (a) Single-mode stationary solution. Steady-state curve of the normalized output field Ixtl as a function of the normalized input field 17for C=5× 104, A= 103, 0~ = 1. The full line curve is obtained from the cubic equation (4.3), the broken curve is obtained from the exact solution (2.12c) using eqs. (4.1), (3.10) and (3.3). (b) The instability domain in the (a, Ixsl ) plane lies above the curve. Same values of the parameters and Yl= 2~±, x,~: ~±. The full line is obtained from the cubic parameter model [see eqs. (4.11 ) ]; the broken curve is obtained from the exact model [see eqs. (2.21 ) ], using also eqs. (4.10) and (2.23).

eqs. (4.5a, b) give the characteristic equation 22"1-22+ 1 + (01 --2 IxSl 2) 2_

ixS 14=0,

(4.6a)

while eqs. (4.5c, d) lead to the eigenvalue equation 22+22+ 1+

(02-21x

s 12) 2 - Ixls 14=0 ;

(4.6b)

note that the two equations differ only in the detuning parameters 01 and 02. The stability of the single-mode stationary solution requires the solutions of eqs. (4.6) to have a negative real part; this is true if and only iJ the constant coefficients o f eqs. (4.6a) and (4.6b) are positive. It is easy to verify that the stability conditio~ from eq. (4.6a), namely 1 + (0~ - 2 I x s 12) 2 - Ix s 1 4 > 0 ,

(4.71

can be written in the form d172/dl x s 1 2 > 0 ,

(4.8',

where the function 172( Ix s 12) is immediately obtained by taking the modulus squared in both m e m b e r s of eq (4.3); hence the stationary state must lie in a positive-slope part o f the steady-state curves shown, for example in figs. 2a, 3a and 4a. Because we want signal generation, we must satisfy the instability condition for the build-up of the side modes 1 + ( 0 2 - 2 1 x s [2)2_ ix s 1 4 < 0 ,

(4.91

which, taking into account eq. (3.12) and introducing the quantity & = (2C/zl 3) I/2ti, gives the condition 378

(4.10

Volume 83, number 5,6 I

I

I

OPTICS COMMUNICATIONS I

I

I

I

I

I

I . . L . i ' ~

15 June 1991

1.2 1.0

2.

I'

0.8 0.6 1. 0.,:; 0.2 0.0

0.0

l i l i l i l i l i l i l i l i l i

0.0

2.

4.

6.

8.

9

10.

0.0

1.0

2.0 IX s)

1

10.

I

0

8.

6.

2.

"-1

0.0 0.0

2.

~.

6.

8.

10.

I xSl ¢' 2 1

&(-)
Fig. 3. (a), (b) Same as figs. 2.a and 2b, but for C=SXI03, ,~= 400, 01= 2; (c) the instabilitydomain from the exact model is shownhere for an extendedrangeof Ixs I.

c~(-+)= [21xs 12-01 ± (IxS 14_ l)Z/2] 1/2

(4.11)

The instability domain in the (&, IxlSl ) plane defined by eqs. (4.11 ) is displayed in figs. 2b, 3b, 4b for the same parameters of figs. 2a, 3a and 4a, respectively. These figures also show the comparison with the instability domain of the exact model (which is the same for model A and model B), prescribed by eqs. (2.21). Note that when the steady-state curve is S-shaped, as in the case of figs. 3a and 4a, the instability domain starts from & = 0, and the extrema of the instability domain for (i= 0 coincide with the boundaries of the negative-slope part of the steady-state curve. Hence, in this case the part of the instability domain in the (dr, IxSl) plane which must be considered for the multimode instability is that for which IxSl > g (see figs. 3b and 4b). Fig. 3c shows the instability domain of the exact model in the (&, IxSl) plane for a large range of values IxSj. By using eq. (2.18), one can show that for J and 0 different from zero the instability domain extends up to values

x/~lxS I ~

(2C/A 3) 1/22C [ (02zJ2 - 1 ) / ( 1 +02) ]1/2 379

Volume 83, number 5,6

OPTICS COMMUNICATIONS

15 June 1991

i

I X~l

i

I

I

I

i

i

1

i

I

i

i

]

I

i

i

l

1

&

3.

3.

2.

////

/,

2.

i

1.

0.0

0.0 0.0

10.

20.

30.

40.

I

0.(

i

I

J

1.

i

2.

3.

4. IXSl ]

Fig. 4. Sameas figs. 2a and 2b, but for C= 5 × 104,A= 103,01= 4. One notes that it is centred around the straight line 6t = q/21xS I, which means that the instability arises for frequencies close to the Rabi frequency of the internal field If s I~ h (i-e., for ~ = x/~ i f s I when 7~= 2yx ). This behaviour is common in many examples of the multimode instability [ 12,15 ]. A characteristic feature of the multimode instability in the limit (3.6) is that it arises specifically from the FWM coupling. As a matter of fact, in the general case of eqs. (2.15 ) this instability may remain possible even in absence of the FWM coupling term 2CT2, thanks to the role played by the 2CTI contribution, which can produce gain. On the contrary, in the limit (3.6), in which eqs. (2.15 ) reduce to eqs. (4.5c, d), 2 CT1 is purely imaginary and therefore does not produce any gain, so that the instability is entirely due to the FWM coupling, which in this limit is represented by the last term in eqs. (4.5a, d). The validity of our parametric model (4.2) is not limited to the neighbourhood of the threshold region, let us now calcul~tte the multimode stationary solutions, by setting dxl/dt' =dx2/dt' =dx3/dt' =0 in eqs. (4.2) with Fx,=O. By using eqs. (4.1), (3.10), (2.3a), (2.6), (2.23) and (2.1a) we see that above threshold the output field displays three components with frequencies

o91, o92=o91-ot(1+ 2Ct¢/7.LA2) , o93=o91+ot(1+2Ctc/7±A2);

(4.12)

note the frequency shifts due to the atomic dispersion. By some algebraic manipulation one obtains that the stationary values of the moduli of the signal fields are equal, Ix31 = Ix21 ;

(4.13)

this result arises from the fact that in the limit (3.6) the absorptive effects are negligible. Furthermore one obtains that in the multimode stationary state the value of Ixl I is linked to that of Ix21 as follows: Ix112=H( Ix212) , H(z)=~{202 - 2 ( 1 + r / ) z + [05 - 3 + (1 +r/)2z2-202(1 +rl)z]l/2}, and finally the modal intensity

Ix212 is linked to

the input intensity by the steady-state equation [46 ]

Y2--41x21211-0102 +402 Ix2 12+01(1 +r/) Ix212-4(1 -I-r/)Ix214 ] + H ( Ix21 E) [1 +02 -20~H( Ix212) +HE( IX2 12) +402 IXE 12--4(4+r/)IX214 ].

(4.15)

The stationary values of the other quantities, i.e. the phase of xl and the sum of the phases of XE and x3 car be calculated easily from the semiclassical equations. The difference between the phases of x2 and x3 is com. 380

Volume 83, number 5,6

OPTICS COMMUNICATIONS

15 June 1991

pletely arbitrary in the multimode stationary state; this phenomenon is well known and is identical to the behaviour of nondegenerate parametric oscillators [47 ]. Before discussing the shape of the multimode stationary solutions described by eqs. (4.13)-(4.15 ), let us briefly remind the steady-state behaviour of the simple model of ref. [23 ], which is obtained from eqs. (4.2) by dropping all phase modulation and cross phase modulation terms. Relation (4.13) is true also for this model, while Ix1 [2 is simply given by the constant value Ix112= (1 +02) 1/2 .

(4.16)

Hence, because Ix~l 2 is given by 172 below threshold and by (4.16) above threshold, the pump intensity Ix, I2 as a function of the input intensity 172 displays, at threshold, a discontinuity of the first derivative. The modal intensity Ix212 obeys the steady-state equation [23 ] 172= (1 +02)1/2{ [ 1 +21x212/(1 +02)1/2] 2+ [01 --202 IX2 ]2/(1 -[-022)1/2] 2} ,

(4.17)

which, by introducing the scaled quantities

9=17/(1-1-02) 1/4 ,

I.~21=x/~lXEI/(l+02)

1/4,

(4.18)

can be written in the form ~/~= (1 +02) [2214+2(1-0102) 1-~2I 2"Jr-1 +02 ,

(4.17' )

which coincides formally with the steady-state equation for the simple degenerate FWM model [23,28 ] and for the degenerate parametric oscillator [48 ]. Hence, using the analysis of ref. [48 ] we can conclude that the steady-state curve of [x212 as a function of the input intensity 172 is monostable for 0102< 1 and bistable for 0102> 1. Furthermore, when the intensity is increased enough above threshold one meets a second threshold, beyond which the multimode stationary solution becomes unstable and the quantities Ix1[2, Ix212 ( = Ix3[ 2) exhibit undamped oscillations; this instability was first predicted in ref. [25 ]. Let us now come back to the complete parametric model (4.2) (neglecting, for the moment, the noise terms Ix, ). The multimode stationary solution governed by eq. (4.15) is substantially more complicated than eq. (4.17 ), especially for the presence of the double sign in the definition of the function H, given by eq. (4.14). This circumstance gives rise to a complex picture, which includes the possibilities predicted by eq. (4.17) but displays several novel possibilities. A complete classification of the various behaviours will be given in a separate publication [49 ]. Here we discuss only two examples, illustrated by figs. 5 and 6, which show the steadystate curve of the fields Ixll and IXEI as a function of 17 obtained from eq. (4.15), for r/= l and the same parameters as in figs. 2 and 3, respectively. These figures indicate also the single-mode stationary solution (broken line) and the jumps among the various branches of the steady-state curve, which occur when the input field is increased or decreased. In the case of figs. 5 (6) the behaviour of Ix2[ at threshold exhibits a second(first-) order phase transition with monostability (bistability) between the multimode stationary solution and the trivial solution x2 = 0, in qualitative accord with the picture predicted by the model of ref. [23 ]. On the other hand, the butterfly hysteresis cycle appearing in fig. 5 for larger values of the input field arises from the double sign in the definition of H and is not possible in the model of ref. [ 23 ]. The same holds for the clockwise hysteresis cycle in fig. 6. For the parameters of fig. 5, there is also an instability that produces undamped oscillations of the moduli. This is shown in fig. 7, which was obtained by numerically solving eqs. (4.2) with a slow sweep (forward and backward) of the input field 17; the shaded regions display the oscillations. The presence of the oscillations is also in qualitative agreement with the simple model of ref. [23]; however, in the case of our parametric model, when the input intensity is increased enough, the oscillations disappear, the butterfly hysteresis cycle is met and finally the system is back to the single-mode stationary solution corresponding to the right boundary of the instability domain of the single-mode steady state. At both upper and lower boundaries of the instability region, the multimode stationary solution merges continuously into the single-mode solution shown in fig. 2a. 381

Volume 83, number 5,6

OPTICS COMMUNICATIONS

15 June 1991

,IJ, i

i

i

i

t

i

i

i

i

I

i

i

i

i

I

i

3.

i

'

I

'

i

i

i

I

'

t

'

t

r

b

3,

IX,~I

I

IX21 / / "

ff

i

2.

2.

1.

I.

0.0

i

I

~

I

O. 0

I

i

L

i

I

10.

I

i

i

I

20.

I

0.0

30.

,

I

O, 0

10.

20.

30.

?

,?

Fig. 5. Steady-state curve from the parametric model (4.2) for (a) the field Ix~l and (b) the field Ix21. The parameters are 0~= 1, 02= 10, )/= 1. The broken line in (a) corresponds to the single-mode stationary solution (fig. 3a ). The arrows indicate the jumps when the input field is increased or decreased. The big arrows on top show the bouadaries of the instability domain of the single-mode stationary solution; at the boundaries of this domain the multimode stationary solution connects itself in a continuous way (but with a discontinuity of the derivative) with the single-mode stationary solution.

2,0

i

i

i

i

l , r , , l l i l l

1.0

I

I

i

'

'

'

I

i

,

~

i

I

b

8

IXll

0.8 1×21 0.6

1.0 0.~

4-

//

_

/ // //

0.2

/ /// i

0.0

0.0

i

,

I

1.

,

i

t

I

2. _ Y

i

,

,

I

3.

Fig. 6. Same as figs. 5a and 5b, but for 0~=

0.0

J

,

0.0

,

I.

2.

?

3.

2, 02 = 3, ~/= 1.

Fig. 8 exhibits the n u m e r i c a l results o f a slow sweep o f the i n p u t field Y, o b t a i n e d from eqs. ( 2 . 2 2 a - h ) o m o d e l B, for values o f the p a r a m e t e r s c o r r e s p o n d i n g to those o f fig. 5. Clearly there is good accord with th~ result o f the p a r a m e t r i c m o d e l shown in fig. 5; the b o u n d a r i e s o f the oscillatory d o m a i n a n d o f the butterfl,. cycle display a shift with respect to fig. 5. In the n u m e r i c a l calculation using m o d e l B, the fields Ix2 i a n d Ix3 are not exactly identical b u t exhibit a small difference on the o r d e r o f 2%. Similarly figs. 9a, 9b (9c, 9 d ) shov a slow sweep o f the i n p u t field o b t a i n e d f r o m the p a r a m e t r i c m o d e l ( m o d e l B) for values o f the p a r a m e t e r corresponding to those o f fig. 6; note that the p a r a m e t e r values o f figs. 9c a n d 9d are c o m p a t i b l e with t h o s o f the e x p e r i m e n t r e p o r t e d in ref. [ 32 ]. Fig. 10 is o b t a i n e d by p e r f o r m i n g a slow sweep o f ]Y with the p a r a m e t r i c m o d e l (4.2), for the same value 382

Volume 83, number 5,6 I

3.

OPTICS COMMUNICATIONS I

I

I

I

I

I

I

i

I

15 June 1991 i

I

a

IXII

i

i

I

i

i

i

i

i

r

i

i

i

I

b

3.

IX21 2.

2.

/ 1.

0.0

I

t

i

0.0

I

J

i

i

J

10.

I

i

~

20.

~

i

I

0.0

30.

?

O. 0

10.

20.

30.

?

Fig. 7. Eqs. (4.2) of the parametric model are solved numerically with a slow forward and backward sweep of the input field 1~. The figures show the behaviour of (a) Ixil and (b) Ix21.Parameters as in fig. 5. r

3.

l

I

I

r

I

I

'

I

i ~ ~ -

I

a

3.

2.

2.

1.

1.

0.0 0.0

,

,

i

~

I

~

10.

~

J

~

I

i

20.

i

?

,

i

I

I

I

I

I

I

I

I

I

I

I

I

I -~

-

b

.i

0.0

30.

I

~

O.

i

i

10.

20.

,

?

,

30.

Fig. 8. The equations of model B are solved numerically with a slow forward and backward sweep of the input field Y. The figures show the behaviour of (a) Ixll and (b) Ix21.The parameters are C=5 X 104, zI= 103, 01= 1, Ym=27j., x=0.017±, t~= 300 (hence 02= 10). o f the parameters o f fig. 5 but for t/=2. The qualitative behaviour is the same as in fig. 5, but there are significant quantitative differences.

5. Spectrum of the fluctuations of the intensity difference between the signal fields above threshold This spectrum is defined by [ 50 ] 4-00

2xNs

S ( o 9 ) = 1 + if~12+lf3l z

d t ' e -'°~c ( ( 6 I _ ( t ' )

6I_(0)),

(5.1)

~oo

where lf2 1 and If3 1 denote the stationary values in a multimode stationary solution, I _ is defined as the intensity difference I_ = f ~f2 - f ~f3,

( 5.2 )

383

Volume 83, number 5,6 2.0

I

l

l

t

OPTICS COMMUNICATIONS

l

l

l

l

l

I

15 June 1991

1.0

t r l l t l

i

r

a

I I ~

II

I

I

--i

b 0.8

IXll IX21

/

/

O.d 1.o

t

O.L;

--4

_1 -i

0.2 0.0

0.0

l l l l t l l l l l l l l l

0.0

1.

2.

~ ,

i

0.0

3.

i

t

t

I

1.

I

t

t

I

3.

2. ?

2.0

i

i

i

i

I

I

i

i

I

I

I

I

i

I

1.0

/

i

i

F

I

i

t

i

i

i

i

i

I

d

0.8

J

2

--

--4

1×21

I×11

O.d 1.o

/

0.,~ 0.2

0.0

0.0 0.0

1.

2.

?

3.

O.

~.

2.

?

3.

Fig. 9. (a), (b) Same as figs. 7a and b, but for the parameters of fig. 6; (c), (d) same as figs. 8a and b, but with C=5X 103, J=400 0t =2, 7M= 2y±, r=0.01yx, ~= 80 (hence 02= 3). The broken lines correspond to the backward sweep. and 8I_ indicates the fluctuation of I_ around the multimode stationary state. It is understood that the tim~ correlation function is calculated using the P-representation. With this definition, the shot-noise level corre sponds to S = 1 and complete suppression Of quantum noise corresponds to S = 0. If we use the parametric model (4.2), the calculation of the spectrum can be performed analytically. On~ reformulates the Langevin equations (4.2) in terms of modulus and phase of the variables f~ (i = 1, 2, 3 ), an( then one linearizes the equations around a multimode stationary solution. T h e linearization is performed onl,. with respect to the moduli, t o the phase off~ and to the sum of the phases off2 and f3, following the samq procedure described in ref. [27 ]. Details will be shown elsewhere [49 ], the result is very simple and read: [461 S(co) = 1 --4K2/(0)2+4/¢ 2) .

(5.3

One notes immediately that eq. (5.3) coincides with the expression of the spectrum first derived for the non degenerate parametric oscillator above threshold [33,34], which predicts complete suppression of quantun noise at zero frequency. This formula holds also for the simple parametric model of ref. [ 23 ], as shown in ret [27 ], and, as we prove here, it remains valid even after including the terms which describe phase modulatio] 384

Volume 83, number 5,6 i

i

i

OPTICS COMMUNICATIONS i

I

i

i

i

i

I

i

i

i

i

15 June 1991

I

i

i

i

i

I

i

i

i

i

I

i

[

i

~

i

I

b

3.

IX21

I X l l 2. 3"

2.

1.

0.0

I

I

I

I

0.0

I

I

I

I

10.

I

I

I

20.

I

I

I

0.0

30.

I 30.

,

0.0

10.

20.

1'

i'

Fig. 10. Same as figs. 7a and 7b, but for t/=2, while the other parameters are unchanged.

and cross phase modulation. Hence expression (5.3) seems to possess a sort of general validity. Note that eq. (5.3) follows from the Langevin equations both for t/= 1 and ~/=2. With the help of eqs. (2.22) of model B we can calculate the spectrum (5.1) also beyond the limit of validity (3.6) of the parametric model (4.2). To this aim, we reformulated eqs. (2.22) in terms of the moduli and phases of the variables f~, Pi, di and we linearized the Langevin equations around a multimode stationary solution. This linearization is performed only in the variables which have a nonarbitrary stationary value. Again, the details of the calculations are described in a separate publication [ 49 ]. The spectrum S(to) is obtained by combining elements of the spectral matrix defined as usual [ 51,52 ] by 2rNs

if212+ if3 i~

[(A+io)l)_lD(AX_iml)_l ]

(5.4)

where A and D are the drift and diffusion matrices, respectively, of the linearized Langevin equations. In the resonant case A= 0, 01 = 0 one meets at most a small reduction of quantum noise [ 53 ] as is shown in fig. 11. This fact is presumably due to linear and nonlinear absorption. On the other hand, fig. 12 exhibits

1.6 .

.

.

.

i

.

.

.

$(e)

i

.

1.2

i'

t,

r,

f,

i,

r,

I,

i , i ,1

1.0

S(e) 1.2

O.d 0.2

1.0 . . . .

-10.

i

,

,

i

i

0.0

0.0

i l l , l , , ,

10.

Fig. 11. The spectrum of the intensity difference between the signal fields, obtained from model B is shown in the resonant case C=20, ~=0, 0=0, t~=8, x<<7±, 7M=2~,±, Y= 16.84.

-4.

-2.

I 0.0

2.

4. (a/k

Fig. 12. Same as fig. 11, but the parametric values are those of figs. 3, 6 and 9, and 1~= 1.69 (broken line). The full line describes the analytic curve given by eq. (5.3).

385

Volume 83, number 5,6

OPTICS COMMUNICATIONS

15 June 1991

the spectrum S(oJ) calculated from model B, for values of the parameters that satisfy conditions (3.6), and are compatible with the values of the experiment of ref. [ 32 ], in which the spectrum was observed. Clearly the curve is everywhere close to that prescribed by the analytic formula (5.3). From the viewpoint of quantum noise reduction there are other quantities of interest, as, for example, the spectra of intensity fluctuations of the single fields f~ ( i = 1, 2, 3), or the squeezing spectra. The calculation of these quantities from the parametric model and from model B will be discussed separately [49 ].

6. Concluding remarks We started our analysis from two different and nonequivalent approximations of the complete multimode model. In both of them we restricted our analysis to three cavity modes, i.e. one pump mode and two signal modes, whose frequencies are symmetrically located around the frequency of the pump mode. In the case of model A we retain all the modal components for the atomic variables, whereas in the case of model B we consider only three components for the atomic polarization and inversion, in analogy with what is done for the field. We formulate precisely a limit [ see eqs. (3.6) ] in which model A and model B reduce exactly to appropriate parametric models, including all the physical effects that are relevant in the limit. These effects are four-wave mixing, phase modulation and cross phase modulation; nonlinear absorption does not play any role because of the large atomic detuning. It is interesting that only the parametric model derived from model A is equivalent to a quantum mechanical master equation [ see eq. (3.16) ], in which the mode-mode interaction is expressed by appropriate hamiltonian terms. It is remarkable that all the stationary solutions of the parametric models can be calculated in closed analytic form. The multimode stationary solutions exhibit a complex phenomenology that displays interesting features in addition to those predicted by the model formulated by Savage and Walls [ 23 ], which does not include the phase modulation and cross phase modulation contributions. Hence our model exhibits not only substantial quantitative changes, but also essential qualitative changes with respect to the model of ref. [23 ]. The comparison of the solutions of model B with those of its parametric approximation in the proper range of parameter values to ensure model validity, is quite satisfactory. This is true both for the semiclassical results and for the spectrum of the fluctuations in the intensity difference between the signal fields, which is the only quantum mechanical quantity that we considered in this paper. Noteworthy is the fact that the parametric models predict for this spectrum the same simple analytic form that was first derived for the case of the optical parametric oscillator above threshold [ 33,34]. Let us now end this paper with some critical remarks. Model A does not lend itself in a straightforward way to the calculation fo the spectrum S(o9) and of the other quantum mechanical quantities. The considerations given in section 3 before eq. (3.16) suggest to approximate model A with model B amended with the inclusion of the inversion amplitudes d2 and d_ 2; it is not obvious, anyway, that this is feasible also beyond the parametric approximation domain. This point is still to be tested. In our analysis we have always considered only three cavity modes. This approximation is reasonable provided that no other mode becomes unstable. This condition can be easily fulfilled, as one realizes immediately by looking at the shape of the instability region given, for example, in fig. 3c. As a matter of fact the instability condition is given by eqs. (2.21) and therefore the instability domain in the plane of the parameters (an, If s I ) is the same for all the modes. From inspection of fig. 3c it is evident that, because of their differenl frequencies, distinct modes become unstable for different values of If s I. This is true also in the parametric approximation, in which the instability domain for the generic mode n is still given by eqs. (4.11 ), where in the definition (4.10) of dt one replaces t~ by the normalized frequency offset of the nth mode t~, = ot,,/?±. A

386

Volume 83, number 5,6

OPTICS COMMUNICATIONS

15 June 1991

description which includes all cavity modes is provided by the full set of eqs. (2.9) or by the MaxweU-Bloch equations (see, e.g., refs. [ 12,15 ] ). The multimode instability predicted by our parametric model [ see eqs. (4.11 ) ] remains unaltered if we consider, instead of a plane wave, a fixed gaussian transverse configuration of the electric field, as is done, for example, in ref. [ 54 ]. As a matter of fact, in this case eqs. (4.2) change only in the sense that all the nonlinear terms are multiplied by the same constant factor. If this constant factor is included in the definition (3.15) of g and in eqs. (4.1), the Langevin equations (4.2) remain completely unchanged. We note that FWM models which include cross-phase modulation between pump and signal fields were considered in refs. [ 55-57 ]. In these models the instability which can lead to the growth of the signal fields occurs only in the negative-slope part of the singlemode steady-state curve (4.3), and therefore nonzero solutions for the sideband intensities are inaccessible [ 57 ]. On the contrary in our case the instability can arise in a positiveslope part of the steady-state curve (4.3), and this feature is a consequence of the difference between the detuning parameters 02 and 01 (see eq. (3.12 )), which follows exactly from our first principle derivation of the model. We observe finally that a further simplification in our models can be obtained by considering the case of a pump field slightly tilted with respect to the axis of the cavity, in such a way that it does not feed mode 1, but interacts directly with the atoms and travels through the medium only once because there is no feedback from the cavity. The situation is similar to that of the experiment in ref. [ 32 ], and is related to the configurations considered in the theoretical treatment of refs. [ 58,59 ]. The advantage of this case is that, because our model assumes otabL << 1, the single-pass effect in the pump field is irrelevant and one can neglect the pump depletion. However, it requires an extremely intense input field because the pump intensity is no longer enhanced by the multiple pass in the cavity. This case is described by our models by treating fl (and, consequently, xl ) as a fixed constant instead of a variable. Precisely, one drops the time evolution equation f o r f l (or xl ) and, in the other equations, one replacesf~ by a constant Y [or 1Tin eqs. (4.2b), (4.2c); in the definition (2.5) of Yone must drop the factor x/~]. Remarkable is the fact that the instability conditions for the build-up of the side modes remain unchanged (i.e., the instability domains shown in figs. 2b, 3b, 4b, are unaltered with IxSl replaced by ]Y). The stationary solution of eqs. (4.2b), (4.2c) is given by eq. (4.13) and

IX212=H-](Y2),

(6.1)

where H - i is the inverse of the function H defined by eq. (4.14) and is explicitly given by

1.2 1.0

IX21 0.8

r

0.6 O.d 0.2 0.0

,

0.0

,

,

,

I

.

.

.

.

1.0

.

.

.

2.0

Fig. 13. Steady-state curve from the simplified model [see eqs. (6.1), (6.2) ], for 02= 3, ~/= 1.

387

Volume 83, number 5,6

H 2 ( z ) = ( 1 + q ) - I [02-2z+_ (z 2 - 1 )I/2] .

OPTICS COMMUNICATIONS

15 June 1991

(6.2)

C u r v e ( 6 . 1 ) is s h o w n in fig. 13 for the s a m e v a l u e s o f the p a r a m e t e r s as i n fig. 6. Expression ( 5 . 3 ) o f the s p e c t r u m S(co) r e m a i n s u n a l t e r e d i n the s i m p l i f i e d model.

Acknowledgements We are grateful to Prof. Walls for s e n d i n g us a copy o f refs. [24 ] a n d [26 ] p r i o r to p u b l i c a t i o n . T h i s research was w o r k e d o u t i n the f r a m e w o r k o f the E S P R I T Basic R e s e a r c h A c t i o n o n Q u a n t u m Noise R e d u c t i o n i n Optical Systems ( N O R O S ) .

References [ 1] B.R. Mollow, Phys. Rev. A 5 (1972) 2217. [ 2 ] F.Y. Wu, S. Ezekiel, M. Ducloy and B.R. Mollow, Phys. Rev. Lett. 38 ( 1977 ) 1077. [3] T. Fu and M. Sargent Ill, Optics Lett. 4 (1979) 366. [4] R.W. Boyd, M.G. Raymer, P. Narum and D.I. Marten, Phys. Rev. A 24 ( 1981 ) 411. [ 5 ] D. Grandclement, G. Grynberg and M. Pinard, Phys. Rev. Lett. 59 (1987) 40, 44. [6] G. Khitrova, J.F. Valley and H.M. Gibbs, Phys. Rev. Left. 60 (1988) 1126. [ 7 ] R. Bonifaeio and L.A. Lugiato, Lett. Nuovo Cimento 21 (1978) 510. [8] L.A. Lugiato, Optics Comm. 33 (1980) 108. [9] R. Bunifacio, M. Gronchi and L.A. Lugiato, Optics Comm. 30 (1979) 129. [ 10] M. Gronchi, V. Benza, L.A. Lugiato, P. Meystre and M. Sargent III, Phys. Rev. A 24 ( 1981 ) 1419. [ 11 ] L.A. Lugiato, V. Benza, L.M. Narducci and J.D. Farina, Z. Phys. B 49 ( 1983 ) 351. [ 12 ] L.A. Lugiato, in: Progress in optics, Vol. XXI, ed. E. Wolf (North-Holland, Amsterdam, 1984) pp. 69 ft. [ 13 ] M.L. Asquini, L.A. Lugiato, H.J. Carmichael and L.M. Narducci, Phys. Rev. A 33 ( 1986 ) 360. [ 14] B. Segard and B. Macke, Phys. Rev. Lett. 60 (1988) 412. [ 15 ] B. Segard, B. Macke, L.A. Lugiato, F. Prati and M. Brambilla, Phys. Rev. A 39 (1989) 703. [ 16] L.A. Lugiato, Z. Phys. B 41 ( 1981 ) 85. [ 17] LA. Lugiato and F. Castelli, Z. Phys. B 64 (1986) 375. [ 18] H.P. Yuen and J.H. Shapiro, Optics Lett. 4 (1979) 334. [ 19] P. Kumar and J.H. Shapiro, Phys. Rev. A 30 (1984) 1568. [ 20 ] M.D. Reid and D.F. Walls, Phys. Rev. A 31 ( 1985 ) 1622; R.E. Siusher, B. Yurke, P. Grangier, A. La Porta, D.F. Walls and M.D. Reid, J. Opt. Soc. Am. B 4 (1987) 1453. [ 21 ] M.D. Reid and D.F. Walls, Phys. Rev. A 34 (1986) 4929. [22 ] D.A. Holm and M. Sargent III, Phys. Rev. A 35 (1987) 2150. [23 ] C.M. Savage and D.F. Walls, J. Opt. Soc. Am. B 4 (1987) 154. [24] W. Zhang and D.F. Walls, Phys. Rev. A 41 (1990) 6385. [25 ] L.A. Lugiato, P. Galatola and L.M. Narducci, Optics Comm. 76 (1990) 276. [26] W. Zhang and D.F. Walls, Optics Comm. 79 (1990) 497. [27 ] B.C. Sanders and M.D. Reid, Phys. Rev. A, in press. [28 ] Wang Kaige and L.A. Lugiato, Ann. Physik, in press. [ 29 ] R.E. Slusher, L.W. Hollberg B. Yurke, J:C. Mertz and J.F. Valley, Phys. Rev. Lett. 55 (1985) 2409. [30] R.M. Shelby, M.D. Levenson, S.H. Perlmuner, R.G. De Voe and D.F. Walls, Phys. Rev. Lett. 57 (1986) 691. [ 31 ] M.W. Maeda, P. Kumar and J.H. Shapiro, Optics Left. 12 (1986) 161. [32] M. Vallet, M. Pinard and G. Grynber~ Europhys. Lett. 11 (1990) 739. [33] A. Heidmann, R.J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre and G. Carny, Phys. Rev. Lett. 59 (1987) 2555. [ 34 ] S. Reynand, C. Fabre and E. Giacobino, J. Opt. Soc. Am. B 4 (1987) 1520. [ 35 ] L.A. Lugiato and L.M. Narducci, in: Fundamental systems in quantum optics, Proc. Les Houches Summer Course, eds. J. Dalibard and J.M. Raimond (North-Holland, Amsterdam) in preparation. [ 36 ] R.J. Glauber, Phys. Rev. 130 ( 1963 ) 2529; 131 (1963) 2766. [37 ] H. Haken, Laser theory, Envyclopaedia der Physik, Vol. XXV/2c, ed. L. Genzel (Springer, Berlin, 1970). 388

Volume 83, number 5,6

OPTICS COMMUNICATIONS

15 June 1991

[38] P.D. Drummond and C.W. Gardiner, J. Phys. A 13 (1980) 2352. [39] R. Bonifacio and L.A. Lugiato, Lett. Nuovo Cimento 21 (1978) 517. [40] S.S. Hassan, P.D. Drummond and D.F. Walls, Optics Comm. 27 (1978) 480. [ 41 ] L.A. Lugiato and L.M. Narducci, Phys. Rev. A 32 ( 1985 ) 1576. [ 42 ] R. Bonifacio, M. Gronchi and L.A. Lugiato, Nuovo Cimento B 53 (1979) 311. [ 43 ] L.A. Lugiato, L.M. Narducci and R. Lefever, in: Lasers and synergetics, eds. R. Graham and A. Wunderlin (Springer, Berlin, 1987 ) pp. 59, 60. [44] P. Grangier, J.F. Roch and G. Roger, in: Proc. ECOOSA Conf. (Rome, November 7-9, 1990), ed. M. Bertolotti (Hilger, Bristol) in press. [45] H.M. Gibbs, S.L. McCall and N.C. Venkatesan, Phys. Rev. Lett. 36 (1976) 113. [46] M. Brambilla, F. Castelli, L.A. Lugiato, F. Prati and G. Strini, in: Proc. ECOOSA Conf. (Rome, November 7-9, 1990), ed. M. Bertoiotti (Hilger, Briston) in press. [47] R. Graham, Z. Phys. 210 (1968) 319; 211 (1968) 469. [ 48 ] L.A. Lugiato, C. Oldano, C. Fabre, E. Giacobino and S. Reynaud, Nuovo Cimento D i 0 (1988) 959. [49] M. BrambiUa, F. Castelli, L.A. Lugiato, F. Prati and G. Strini, in preparation. [ 50 ] A.S. Lane, M.D. Reid and D.F. Walls, Phys. Rev. A 38 (1988) 788. [51 ] B. Yurke, Phys. Rev. A 29 (1984) 408. [ 52 ] M.J. Collett and C.W. Gardiner, Phys. Rev. A 30 (1984) 1386; M.J. Co]lett and D.F. Walls, Phys. Rev. A 32 (1985) 2887. [ 53] M. Brambilla, G. Strini, F. Castelli, L.A. Lugiato and F. Prati, in: Proc. Conf. Nonlinear dynamics of optical systems (Afton, June 4-8, 1990), eds. N. Abraham, E. Garmire and P. Mandel (Opt. Soc. Am.) in press. [54] L.A. Lugiato and M. Milani, Z. Phys. B 50 (1983) 171; L.A. Lugiato, R.J. Horowicz, G. Strini and L.M. Narducci, Phys. Rev. A 30 (1984) 1366; L.A. Orozco, H.J. Kimble, A.T. Rosenberger, L.A. Lugiato, M.L. Asquini, M. Brambilla and L.M. Narducci, Phys. Rev. A 39 (1989) 1235. [ 55 ] G.J. Milburn, D.F. Walls and M.D. Levenson, J. Opt. Soc. Am. B I (1984) 390. [56] R.M. Shelby, M.D. Levenson, D.F. Walls, A. Aspect and G.J. Milburn, Phys. Rev. A 33 (1986) 4008. [ 57 ] G.J. Milburn, M.D. Levenson, R.M. Shelby, S.H. Perlmutter, R.G. DeVoe and D.F. Walls, J. Opt. Soc. Am. B 4 (1987) 1476. [58] F.A.M. de Oliveira and P.L. Knight, Phys. Rev. Lett. 61 (1988) 830. [59] G.S. Agarwal, Optics Comm. 80 (1990) 37.

389