Phase instability in two-photon optical bistability

Volume 69, number 5,6

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15 January 1989

PHASE INSTABILITY IN T W O - P H O T O N O P T I C A L B I S T A B I L I T Y

P. GALATOLA, L.A. L U G I A T O , M. V A D A C C H I N O Dipartimento di Fisica del Politecnico, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

and N.B. A B R A H A M 1 Institute for Scientific Interchange, Villa Gualino, Viale Settimio Severo 65, 10133 Torino, Ita(v

Received 27 September 1988

We analyze the problem of single-mode two-photon optical bistability using a model which neglects the intensity-dependent Stark shift. We demonstrate the existence of a phase instability which arises in bad-cavity conditions in the lower branch of the steady-state curve, and study the spontaneous oscillations which emerge from this instability.

The problem o f squeezed state generation in nonlinear optical systems is the subject of increasing attention both from theoreticians and from experimentalists, especially after the first experimental observations of this phenomenon [ 1 ]. One-photon optical bistability (OB) with two-level atoms is an example of a system in which this goal has been achieved. After the first prediction [ 2 ], this problem was analyzed theoretically in ref. [3], and the experimental observation was obtained at the University of Texas at Austin [4]. Since the very beginning o f the theoretical research on squeezing in OB it was clear, however, that two-photon OB provides a more convenient framework to obtain a large level of squeezing [5 ]. Ref. [5 ] analyzed this effect in the intracavity field, for the purely absorptive case in the good-cavity limit. The spectrum of squeezing in the output field was calculated in ref. [6 ] both for a A medium model and for a ladder medium model. Even if the treatment allows for arbitrary values of the ratios of the atomic relaxation rates to the cavity damping constant, the analysis for the ladder case is focused on the good-cavity limit. Ref. [7] studies squeezing in Permanent address: Department of Physics, Bryn Mawr College, Bryn Mawr, PA 19010, USA 414

two-photon OB including the presence of the intensity-dependent Stark shift, again in the good-cavity limit. From a semiclassical viewpoint, a complete theory for two-photon amplifiers and absorbers was first developed in ref. [ 8 ], which reduces the problem to an effective two-level model. In order to perform a quantum statistical treatment, in our work we neglect the intensity-dependent Stark shift discovered in the analysis of ref. [ 8 ]. This step is exact only for special selections of the physical parameters, however we prefer to start our analysis from the simplest situation. Thus, the single-mode model that we consider coincides with that used in refs. [5] and [6] (ladder medium version). We focus mainly on the bad-cavity situation, which was shown to be most convenient in the one-photon case [ 9 ]. In this paper we prove the existence of a phase instability which arises in a segment of the steady-state curve of output intensity as a function of input intensity in badcavity conditions, and we study the spontaneous oscillations that emerge from the phase instability. In the companion article ref. [10] we analyze the effects that the presence of this phase instability has on the spectrum of squeezing of the output field. The starting point of our analysis is the master equation given in ref. [ 5 ], generalized to include the

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15 January 1989

possibility of atomic and cavity detunings. This is identical to the master equation of the one-photon case [ 9 ], with the exception of the commutator with the interaction hamiltonian which has now the form

J ? = # [ - (1 + i O ) x + y + 2 a C x * v ] ,

(7a)

i,= ( 1 / 2 f ) [ - (1 +iA)v+x2m] ,

(7b)

rh= - ½[ (x*)2v+x2v * ] - m + 1 ,

(7c)

HAy =ihg [ (a t) 2R - a Z R + ] ,

where the time is normalized to 7~ ~, x is the output field normalized to the square root of the saturation photon number, v and m are normalized variables which correspond to the atomic polarization and population difference, respectively. The quantities x* and v* obey to the complex conjugates ofeqs. (7a) and (7b), respectively. The semiclassical steady-state equation reads

-

( 1)

where g is the effective coupling constant, a and a* are the annihilation and creation operators of photons of the single cavity mode considered in the model, and the operators R - and R + are associated with the macroscopic polarization of the effective two-level medium. The independent parameters of the model are the ratio -

/z = x/71i,

(2)

of the cavity damping constant x ( = c a v i t y linewidth) to the relaxation rate 7, of the population inversion; the ratio -

f = 711/271 ,

(3)

of y, to twice the relaxation rate 7__ ( = atomic linewidth) of the atomic polarization; the condition of purely radiative damping corresponds to f = 1; the bistability parameter

lYl = Ixl

+[Oar

1

1~ 1 4 _ ]

2°'CAIxl2

12~ '/2

l~t:i~4j

j

.

(8)

One can easily verify that we can always select the phase of y, which is arbitrary, in such a way that the stationary value of x is real and positive; with this choice in the resonant case, A = O= 0, the steady-state equation reduces to y = x [ 1 - 2 6 C x 2 / ( 1 -I-x4 ) ] ,

(9)

-

C= (gN/2x)

(YlI/Y±)1/2,

(4)

where N is the number of atoms; the cavity detuning parameter -

O= ( 09c -09o)/X,

(5)

where 09c and 09o are the frequencies of the cavity mode and of the input field, respectively; the atomic detuning parameter -

A= ((.oa -209o)/y± ,

(6)

where 09a is the atomic transition frequency; the parameter y which indicates the input field normalized to the square root of the saturation photon number (y,y±) ~/2/2g;, the unsaturated inversion per atom - 1 ~
-

and therefore y is also real. In the case of OB ( a = - 1 ) the steady-state curve of x versus y given by (9) is S-shaped for C>2.71 [11]. The stability analysis of eqs. (7) linearized around a stationary solution leads to a fifth order algebraic equation for the eigenvalues of the linearized problem. The constant term of this equation is proportional to d lYl 2/d Ixl 2, and therefore ensures the usual instability of the negative-slope portion (if any) of the steady-state curve. This instability corresponds to the fact that a real eigenvalue vanishes in correspondence to the turning points of the steady-state curve. On the other hand, in suitable domains of the parameter space one meets also an instability arising from a Hopfbifurcation, such that a pair of complex conjugate eigenvalues has a real part that vanishes in correspondence to the boundary of these domains. This type of instability leads to the rise of spontaneous oscillations in the output intensity. The linear stability problem can be treated analytically in the resonant case A = 0 = 0 , in which the 5 × 5 matrix of the linearized problem splits into a 3 X 3 matrix for 415

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the a m p l i t u d e fluctuations and a 2 × 2 matrix for the phase fluctuations. The 3 × 3 m a t r i x was analyzed in ref. [ 12 ] and does not lead to the onset o f oscillatory instabilities in the case o f OB. In contrast, in the case of the laser with injected signal the a m p l i t u d e fluctuations produce oscillatory instabilities for appropriate selections o f the parameters, but we do not discuss t h e m here. The 2 × 2 matrix leads to the eigenvalue equation 22+ [ 1/ ( 2 f ) + 1~(2--y/x) ]2 -I- ( / t / 2 f ) ( y / x ) = O ,

(I0)

where y / x is explicitly given by eq. ( 9 ) and the eigenvalue 2 has been n o r m a l i z e d to y,. The phase instability arises when the coefficient o f 2 in eq. ( 1 0 ) becomes negative, i.e. for

y/x>2+l/(21zf).

15 January 1989

Hence the instability arises in the segment o f the steady-state curve defined by ( 1 2 a ) p r o v i d e d the condition C > l + ( 2 / t f ) -~ is satisfied. F r o m eqs. (10) and ( 11 ) the frequency o f the spontaneous oscillations at the instability threshold is given by .Q~ = [ ( / z / 2 f ) (2 + 1 / ( 2 / z f ) ) ] 1/2--+ (.0c ~,-,~~)li~c

= [ (Kyl,/2f) (2 + 1 / (2/~f)) ] 1/2

( 13 )

This instability can arise also in absence o f bistability; when the steady-state curve o f x versus y is Sshaped, the lower instability threshold x lies in the

0.07

(11)

C o n d i t i o n ( 11 ) cannot be satisfied in the case o f a laser with injected signal, because y / x is always smaller than unity. In contrast, in the case o f OB ( a = - 1 ) using ( 9 ) one sees that eq. ( 11 ) is fulfilled for [C-(C 2- 1)1/2]'/2=x -
0.06

< x + = [ C + ( C 2 - 1 )1/2] 1/2, C = C [ 1+ ( 2 # f ) - ' ]

ixl 2

l i

(12a)

l i

I

t l

0

5

(12b)

-'

I

I

I

I

I

1

[

,

I

i

I

J

'

I

i

i

I

i

I

I

j

I

I

I

J

i

l

I

0,1

o

Imx

0

X ""',.. -o.1

(b) I

.f.

( ,

t

.

I

,

I

,

I

.

I

.

I

.

I

.

I

,

l

.

I

.

I

,

I

10

Fig. 1. Steady-state curve of output field x as a function of input field y for C= 10, A=0, O=0, a= - 1. The broken part indicates the unstable portion of the steady-state curve for f = 1, #= 5; the lower instability threshold is for y~ 0.5. 416

L

i

0.24

I

I

i

ReX

'.25

Fig. 2. Spontaneous oscillations for C= 10, A=0, 0=0, 6= - 1, #= 5,f= 1, y=0.516. (a) The long term evolution of the output intensity. (b) The projection of the phase space trajectory onto the plane (Re x, Im x).

Volume 69, number 5,6

OPTICS COMMUNICATIONS

lower branch, whereas the upper instability threshold x+ can lie in the upper branch or, as in the case of fig. 1, in the negative slope branch. Note that in the good-cavity direction (/t << 1 ) this instability can occur only when C is very large, whereas for f,~ 1 in the bad-cavity direction (/t >> 1 ) the instability occurs also for moderate values of C. In the latter limit, the instability condition ( 11 ) becomes y > 2x, C coincides with C and the instability arises for C > 1. For the parameters of fig. 1, which corresponds to the case o f O B ( a = - 1 ), numerical solutions of eqs. (7) show that in this case at the instability threshold x_ = 0.235 ( y = 0.493 ) the system displays a forward H o p f bifurcation with the appearance of a stable

I

. . . .

i

15 January 1989

small amplitude limit cycle. At its inception the instability is a phase modulation with periodic oscillation of their imaginary component o f the field. The intensity pulsations are much weaker and occur at twice the frequency of the phase pulsations. The amplitude of the spontaneous oscillations grows under an increase of the input field. Fig. 2 shows the spontaneous oscillations, and the projection of the limit cycle onto the plane of variables Re x, Im x, for a value of the input field slightly above the critical value for instability. The behavior of the output intensity when y is increased may induce to the erroneous identification of a period doubling (fig. 3a); in fact, this phenomenon arises from the distortions of the . . . .

. . . .

i

. . . .

[

. . . .

~

. . . .

[

(a)

1

10

Ixl 2

ixl 2

i , l =

i

I

i

i

L

~

,

,

I

,

,

o

L

h

Yi, t

0

1

' '

'

'

I

' ' i

li

i

r

I

I

i'

i

I

I

'

~

. . . .

I

. . . .

r

. . . .

I

I

(b)

[ i X

'

Yiit

'

'

'

(b) Imx

2

-2 -1

l i

i

t

I 0

t

i

t

i

I

=

~

I

I

I 2

=

t

J

t

I

I

I

I

l

l

,

l

l

l

L

l

l

l

o

ReX

Fig. 3. Same as fig. 2, but for y=6.232.

Fig. 4. Same as fig. 2, but for C = f = l , ly[ = 1.611.

l

h

l

,

[

,

l

l

1

Rex

160, A = 20, 0 = 1, or= - 1 , / t = 5,

417

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OPTICS COMMUNICATIONS

lobes o f the initially figure-8 phase space trajectory (fig. 3b); a single p e r i o d o v e r the t r a j e c t o r y n o w corr e s p o n d s to f o u r intensity pulses instead o f the earlier two intensity pulses. T h e oscillations persist w h e n the i n p u t field is i n c r e a s e d up to y ~ 18.2 (this is bey o n d the t u r n i n g p o i n t o f the steady state solution, i n d i c a t i n g a region o f c o e x i s t e n c e b e t w e e n a stable u p p e r b r a n c h and a pulsing s o l u t i o n ) . W h e n y is increased further, the oscillatory a t t r a c t o r b e c o m e s unstable a n d the system j u m p s to the stable s t a t i o n a r y state in the u p p e r branch. Fig. 4 shows an e x a m p l e o f the s p o n t a n e o u s oscillations u n d e r d e t u n e d c o n d i t i o n s ; again, the " p e riod f o u r " structure arises f r o m the c o n f i g u r a t i o n o f the phase space trajectory, and in fact f o u r oscillations o f the o u t p u t i n t e n s i t y c o r r e s p o n d to a single p e r i o d o v e r the trajectory. In a future p u b l i c a t i o n we will s h o w that the phase instability persists w h e n the i n t e n s i t y - d e p e n d e n t Stark shifts [8 ] are i n c l u d e d .

Acknowledgements We are grateful to L.M. N a r d u c c i a n d P. M a n d e l for e n l i g h t e n i n g discussions. T h i s w o r k was c o m pleted in the f r a m e w o r k o f the E E C t w i n n i n g project on N o n c l a s s i c a l States o f the E l e c t r o m a g n e t i c Field and D y n a m i c s o f N o n l i n e a r O p t i c a l Systems.

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15 January 1989

References [ 1 ] See for example the collection of papers contained in J. Mod. Opt. 34 June/July 1987, eds. R. Loudon and P.L. Knight: and J. Opt. Soc. Am. B 4 October 1987, eds. H.J. Kimble and D.F. Walls. [2] L.A. Lugiato and G. Strini, Optics Comm. 41 (1982) 67. [3]D.F. Walls and G.J. Milburn, in: Quantum optics, gravitation and measurement theory, eds. P. Meystre and M.O. Scully (Plenum Press, New York 1983) p. 209: M.D. Reid and D.F. Walls, Phys. Rev. A 32 (1985) 396: M.D. Reid, A. Lane and D.F. Walls, in: Quantum Optics IV, eds. J.D. Harvey and D.F. Walls (Springer-Verlag, Berlin 1986) p. 31; D.A. Holm and M. Sargent III, Phys. Rev. A 35 ( 1987 ) 2150; M.D. Reid, Phys. Rev. A 37 (1988) 4792. [4] M.G. Raizen, L.A. Orozco, Min Xiao, T.L. Boyd and H.J. Kimble, Phys. Rev. Lett. 59 (1987) 198; L.A. Orozco, M.G. Raizen, Min Xiao, R.J. Brecha and H.J. Kimble, J. Opt. Soc. Am. B 4 ( 1987 ) 1490. [5] L.A. Lugiato and G. Strini, Optics Comm. 41 (1982) 374. [6] C.M. Savage and D.F. Walls, Phys. Rev. A 33 (1986) 3282. [7] B.A. Capron, D.A. Holm and M. Sargent Ill, Phys. Rev. A 35 (1987) 3388. [ 8 ]L.M. Narducci, W.W. Eidson, P. Furcinitti and D.C. Eteson, Phys. Rev. A 16 (1977) 1665. [9] F. Castelli, E.A. Lugiato and M. Vadacchino, Nuovo Cimento D 10 (1988) 183: L.A. Lugiato, M. Vadacchino and F. Castelli, Proc. Conf. on Squeezed and nonclassical light, Cortina (Italy), January 1988, eds. R. Picke and P. Tombesi (Plenum Press, N.Y.) in press. [10]P. Galatola, L.A. Lugiato, M. Vadacchino and N.B. Abraham, Optics Comm. 69 (1989) 419. [ 11 ] F.T. Arecchi and A. Politi, Left. Nuovo Cimento 23 ( 1978 ) 65. [ 12 ] S. Ovadia and M. Sargent Ill, Optics Comm. 49 ( 1984 ) 447.