Squeezing in two-photon optical bistability and laser with injected signal

Volume 69, number 5,6

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15 January 1989

SQUEEZING 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 AND L A S E R W I T H I N J E C T E D S I G N A L P. G A L A T O L A , 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 i Institute for Scientific Interchange, Villa Gualino, Viale Settimio Severo 65, 10133 Torino, Italy

Received 27 September 1988

We show that, on approaching the threshold for the phase instability of two-photon optical bistability, the spectrum of squeezing exhibits the onset of a large squeezing on a frequency region on the order of the cavity linewidth. The squeezing becomes perfect in the bad-cavity limit. This phenomenon does not require detuning nor a large value of the bistability parameter, and occurs for values of the input intensity well below the saturation level, and much smaller than that corresponding to the turning point of the steady-state curve. In the resonant configuration the best squeezing arises in the amplitude quadrature component and is therefore accompanied by a strongly subpoissonian photon statistics. We show that in the bad-cavity limit the two-photon laser with injected signal exhibits a significant squeezing in the phase quadrature component.

In a c o m p a n i o n paper [ 1 ], to be referred hereafter as I, we analyzed the semiclassical equations for single m o d e two-photon optical bistability ( O B ) and laser with injected signal, neglecting the intensity-dep e n d e n t Stark shift p o i n t e d out by N a r d u c c i and collaborators [ 2 ]. Our analysis discovered the existence of a phase instability which leads to the onset o f spontaneous oscillations. This instability arises in bad-cavity conditions, i.e. when the quantity 2/~f which represents the ratio o f the cavity linewidth to the atomic linewidth, is substantially larger than unity. In the study o f squeezed state generation, there is an increasing a p p r e c i a t i o n that strong squeezing can be o b t a i n e d just below instability thresholds. This was first noted by Collett and Walls [ 3 ] who found perfect squeezing at critical points (both steady bifurcations, turning points, and H o p f bifurcations). This link has been observed in the recent intracavity Permanent address: Department of Physics, Bryn Mawr College, Bryn Mawr, PA 19010, USA.

second h a r m o n i c generation experiment o f K i m b l e and collaborators [4 ]. We find a similar link between instabilities and strong squeezing in the two-photon OB p r o b l e m discussed here. On approaching the lower threshold o f the phase instability by increasing the input intensity, the output field exhibits large squeezing over a very b r o a d region o f frequency. This effect can be o b t a i n e d also for m o d e r a t e values o f the bistability p a r a m e t e r under resonance conditions and for values o f the input intensity much smaller than the saturation level. Throughout this paper we will use the same notations and symbols o f I. F o r each quadrature component o f the electromagnetic field a~ = a e x p ( - i ~ 0 ) + a*exp(i~0) ,

(1)

where ~0 is an arbitrary phase, the spectrum o f squeezing S(~o, ~0) in the output field is defined as

[51

0 0 3 0 - 4 0 1 8 / 8 9 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )

419

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15January1989

dtexp(-io~t)<:~a~(t) 5a~(0):) ,

S(~o, ~0)=2x j

(2) where 8a~ indicates the fluctuation a r o u n d a stable steady-state value, and : : means n o r m a l ordering. With the numerical factor in eq. ( 2 ) , one has perfect squeezing at frequency ~o when S(oJ, q~) = - 1. The procedure we follow to calculate the function S is exactly that described in detail in ref. [6], after replacement of the one-photon interaction hamiltonian by the two-photon operator given by eq. ( 1 ) o f I. Let us consider first the resonant case C = 1 0 , L/=0=0, a=-1 ( O B ) , f = 1, in which the steadystate curve has the configuration shown in fig. 1. F o r p = 10 -4 (good-cavity l i m i t ) there is no phase instability and one has squeezing over the segment o f the lower transmission branch 0 ~
10

' i , r ' i , i , i ' i ' t ' i ' i ' I ~

x

,r

,

/ _

,-7-,

0

i

,

t

~

I

,

I

,

I

,

I

,

_

I

,

-

I

-

-

,

-

I

10

,

.

.

I

.

.

,

.

I

,

I

,

y Fig. 1. Steady-state curve of output field x as a function of input field y for C= 10, A=0, 0=0, (7= - I. The broken part indicates the unstable portion of the steady-state curve for f= 1, l~= 500. 420

_

0.23

0 -1 15

Fig. 2. ,~is the spectrum of squeezing maximized with respect to the quadrature phase q~; in the resonant case A=0=0 the optimum squeezing occurs for ~0=0. The parameters are C= 10, A= 0= 0, a= - 1,f- 1,/t = 10. The diagram exhibits the variation of the spectrum of squeezing, changed of sign, along the stable segment of the lower transmission branch. The maximum value of -~qis 0.965. Here and in the following figs. 3 and 4, .(2 is the frequency normalized to the relaxation rate 7~ of the population difference. attains a factor o f squeezing larger than 103. Because the best squeezing occurs in the a m p l i t u d e component, as it is true in general in the resonant case A = 0 = 0 , the photon statistics is strongly subpoissonian over a wide region of frequency. We note that the instability threshold is for x ~ = 5 × 10 -2, hence the intracavity intensity (as well as the input intensity) is well below the saturation level when the large squeezing arises. Fig. 4 exhibits the variation w i t h / t o f ( a ) the maxi m u m o f (I + S ) ~ with respect to frequency at instability threshold, ( b ) the value o f the frequency ( n o r m a l i z e d to 71i) which corresponds to the maxim u m squeezing. One sees that (a) the m a x i m u m increases linearly with ~t, ( b ) the m a x i m u m squeezing frequency increases a s / t ~12 and follows basically the behavior o f the instability frequency ~2c (see eq. ( 13 ) o f I), even if it does not coincide with it. In the resonant case ~ = 0 = 0 one obtains the following expression for the spectrum S(~o, q~=0) in the bad-cavity limit:

S(~o,~o=O)=-4(y/x-l + corrections,

)[(y/x)2+(oJ/Jc)2]

' (3 )

Volume 69, number 5,6

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15 January 1989 i

t IIIHq

(a)

i JrlHq

i

i iiiHq

~

J ~llHI + O

10 5

1

10 2

(1.,~)"1 10 4

Qm 10 3

0.23

10

+ 10 2

(

0 -2

[]

i

L ililiii

10 2

f/

~ iiiiiJI

i 10 3

ttliHd

i 10 4

ii~l

la

Fig. 4. Variation of the spectrum of squeezing w i t h / t when parameters are C=10, A = 0 = 0 , ~ r = - l, f = l . ( R ) maximum of ( 1 +,~) ~with respect to the frequency .(2 and to the normalized

200

field x, left hand vertical axis; ( + ) frequency(2m (in units 71~) of maximum squeezing, right hand vertical axis. All axes are on a logarithmic scale.

(b)

1.5.10 3

(1 • ~)-1 0.23

1

-lC

t3

lOO

Fig. 3. (a) Same as fig. 2, but for p=500. (b) Behavior of the quantity ( 1 + S ) - ] ; its peak value is 1.36× 103.

where, in accord with the steady-state equation in resonance, y / x = 1 -- ( 2 a C x 2 ) / ( l

+X4),

(4)

and a = - 1 for OB. For/~--,oo the correction is nonnegligible only for frequencies o9~> 1). In the badcavity limit/1-, oo, at the instability threshold y / x = 2 the dominant term gives perfect squeezing for o9/ x = 0 . The corrections give rise to a dip in the func-

tion -S(o9, ~0=0), centered at o9=0, which shifts the position of the maximum squeezing frequency from 09=0 to a value oc (xT,) ~/2 so that o9/7,oc/~ ~/2. Note that this picture holds for any value of the bistability parameter such that there is a phase instability in the limit/t--,~, i.e. for C> 1, and holds for any value o f f = 7,/27±. The inclusion of detuning changes the picture substantially only in the good-cavity limit, permitting a much larger degree of squeezing than in the resonant configuration. Especially, in ref. [ 8 ] it is shown that in the dispersive limit A>> 1 one attains ideal squeezing at the turning points of the steady-state curve; it must be kept in mind, however, that the bistability parameter C must be much increased in this limit in order to maintain reasonably small the input power necessary to reach the turning point. On the other hand, in the bad-cavity limit the presence of detuning does not change in any way the resonant picture. The analysis of the eigenstates, which correspond to the pair of complex conjugate eigenvalues that produce the instability, shows that the direction of optimum squeezing is essentially orthogonal to the direction of minimal damping-maximal fluctuations indicated by the eigenstates, that is, the squeezing is essentially orthogonal to the direction of the incipient instability. As it will be explicitly shown in a separate publication, in the bad-cavity limit one de421

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rives the following formula for the spectrum of squeezing, which generalises eq. (3):

S(0), tfi)=4ps[ ( l + O 2 - p ~ - t f ) 2 ) 2 +4t£)2]

'

X {2p~ - [ ( 1 - L 9 2 + p ~ ) 2 + 4L92 ] ,/2 - 0 ) 2 ( 1 - 0 2 + p ~ ) [ ( 1 - 0 2 + p ~ ) 2 + 4 0 2 1 -'/2} +corrections,

(5)

where d)=0)/K, and the phase f) is selected in such a way that the degree of squeezing in the dominant contribution to the spectrum, characterized by a width proportional to the cavity linewidth x, is optimized for 0)=0. The quantity Ps is defined as

ps=2CIxI2(I+A2)J/2/(I+A2+Ix[4),

Of course, our analysis is based on a linearized treatment, which becomes invalid in the close neighbourhood of the instability threshold; this feature prevents, as is usual the case, the achievement of complete squeezing. An experimental observation of both the phase instability and the related squeezing would be of outstanding interest and would be equivalent to the phase instability and squeezing observed in the second harmonic generation problem [4]. We remark r

I

i

I

i

I

I

I

I

I

i

I

(a)

~

J

(6)

where x is the steady-state value of the output field. In the limit/~-~ oe the phase instability threshold corresponds to p~= ( 1 + 0 2 ) 1/2, a condition that in the resonant case A = 0 = 0 of OB coincides with the equality y/x= 1+ 2 C x 2 / ( 1 + x 4) - - 2 , that we used before. One verifies easily that at instability threshold the expression (5) reduces t o - 4 / ( 4 + o 3 2 ) , giving again perfect squeezing for 0)=0. In conclusion, two-photon OB presents itself as a competitive candidate to produce large squeezing. The behavior of the spectrum of squeezing is very different from the one-photon case studied in refs. [ 9,10,11,6 ]. Especially, in the latter case large values of the bistability parameter C and of the atomic detuning A are necessary to obtain a significant level of squeezing. The difference is introduced by the presence of a phase instability in the two-photon case, similar to that of second harmonic generation [ 12]. This fact allows for a very simple recipe to obtain a large, squeezing in two-photon OB in the bad-cavity direction: increase the input field to the instability threshold, which corresponds to an input intensity smaller than the saturation level ( lyL 2< 1 ) and much smaller than the intensity at the turning points of the steady-state curve. This procedure does not require large values of the bistability parameter nor of the detunings, and as an extra bonus the width of the spectrum of squeezing scales as the cavity linewidth, which is much larger than the atomic one in the badcavity direction. The ratio of the input to output power at instability threshold is on the order of 4 in the resonant case, which is quite a reasonable figure. 422

15 January 1989

i

I

i

I

0.5

i

l

l

l

[

l

l

I

i

y

F i l l

I I

(b)

-g

o

a

Fig. 5 (a) Steady-state solution of the output field x as a function of input field y for the two-photon laser with injected signal with C=0.6, a = 1, d = O = 0 . The broken part indicates the unstable portion of the curve for f = 1, ~t= 500. (b) Spectrum of squeezing -~q of the phase quadrature component; d~ is the frequency normalized to the cavity linewidth x. The parameters are the same as in (a) with y = 0 . 4 , x = 1, corresponding to the point A of the steady-state curve.

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in this connection that two-photon OB has been observed in rubidium [ 13 ]. We note that the single-mode phase instability discovered here implies the existence of a corresponding multimode instability, as a consequence of the general correspondence principles of ref. [ 14 ] which hold also for the two-photon case. In a future publication, we will show that the phase instability, and the rise of ideal squeezing in approaching the instability threshold in the bad-cavity limit, persist when the intensity-dependent Stark shifts [2] are included. We will also show that the formula (5) is a general result for two-photon processes. For example it coincides with the expression of the spectrum of squeezing for the low frequency mode in degenerate parametric amplifiers below threshold [ 15 ], and in this case the correction is exactly zero. Furthermore, the formula (5) describes the spectrum of squeezing for the low frequency mode in second harmonic generation in the limit in which the high frequency mode is much more strongly damped than the low frequency mode, and in this sense generalises the demonstration of ideal squeezing in the approach to the phase instability threshold, obtained in ref. [3] for the resonant case. For both degenerate parametric amplifiers and second harmonic generation, O in eq. (5) denotes the detuning of the low frequency mode normalized to its damping rate 7~, 6J=oJ/7~ and, using the notations of refs. [9] and [16], one hasps= IA2sl, where A2s is the stationary value of the normalized amplitude of the high frequency mode; the instability boundary corresponds, in both cases, again to the prescription

15 January 1989

prescribes that y/x < 1 in the case of the LIS, eq. (7) shows that there is squeezing over all stable segments of the steady-state curve, at least for frequencies ~o>>7~. The points such that y/x=O (with C~> 1 ), for which eq. ( 7 ) would give ideal squeezing for ~o/ x = 0 , lie in unstable regions. This does not prevent, however, to achieve a significant level of squeezing as shown, for example, in fig. 5b. The possibility of squeezing in a system with population inversion, as the LIS, is quite interesting. A small degree of squeezing in the amplitude quadrature component was predicted for the one-photon LIS [ 17 ], whereas the two-photon LIS does not produce squeezing in the good-cavity limit [7]. We have shown here that the two-photon LIS can give rise to a sizable squeezing in the bad-cavity limit, in the phase quadrature component, even for values of the gain C smaller than the oscillation threshold C= 1 of the two-photon laser (see fig. 5b). That this is a phase squeezing is related to its being orthogonal to the direction of the incipient instability, which is in this case an amplitude instability. We remark in this connection that a recent work [18] showed the possibility of a significant phase squeezing in a special type of LIS, called two-photon correlated spontaneous emission laser. We are grateful to E. Oiacobino, C. Fabre, G. Leuchs, P. Grangier and S. Reynaud for enlightening discussions. This work was completed in the framework of the EEC twinning project on Nonclassical States of the Electromagnetic Field and Dynamics of Nonlinear Optical Systems.

p s = (1 + t 9 2 ) I/2

Eqs. (3) and (5) hold, of course, also for the twophoton laser with injected signal (LIS, ~r= 1 ). For the sake of simplicity let us focus on the resonant case zl=t~=0. In the bad-cavity limit one obtains for the quadrature component of the phase the following expression of the spectrum of squeezing: S(~o, ~/2) = - 4 ( 1- y / x ) + corrections.

[(2-y/x)2+ (og/x) 2 ] - ' (7)

This equation holds also for OB; in the case of the LIS, for which ~ = ~/2, it is a special case of eq. (5). Because the steady-state equation on resonance (4)

References [ I ] P . Galatola, L.A. Lugiato, M. Vadacchino and N.B. Abraham, Optics Comm. 69 (1989) 414. [2] L.M. Narducci, W.W. Eidson, P. Furcinitti and D.C. Eteson, Phys. Rev. A 16 (1977) 1665. [3] M.J. Collen and D.F. Walls, Phys. Rev. A 32 ( 1985 ) 2887. [4] H.J. Kimble, S. Pereira and Min Xiao, Technical Digest of the XVI International Quantum Electronics Conference, paper WB-1, p. 466. [5] B. Yurke, Phys. Rev. A 29 408 ( 1984); M.J. Collet and C.W. Gardiner, Phys. Rev. A 30 (1984) 1386.

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[6]F. Castelli, L.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. [7] L.A. Lugiato and G. Strini, Optics Comm. 41 (1982) 374. [8 ] C.M. Savage and D.F. Walls, Phys. Rev. A 33 (1986) 3282. [9] L.A. Lugiato and G. Strini, Optics Comm. 41 (1982) 67. [10]D.F. Wails 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.

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[ 11 ] 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. [ 12] P.D. Drummond, K.J. McNeil and D.F. Walls, Optica Acta 27 (1980) 321; P. Mandel and T. Erneux, Optica Acta 29 (1982) 7. [ 13 ] E. Giacobino M. Devaud, F. Biraben and G. Grynberg, Phys. Rev. Lett. 45 (1980) 434. [14] L.A. Lugiato and L.M. Narducci, Phys. Rev. A 32 (1985) 1576. [ 15 ] C.M. Savage and D.F. Walls, J. Opt. Soc. Am. B 4 ( 1987 ) 1514. [16] L.A. Lugiato, G. Strini and F. De Martini, Optics Lett. 8 (1983) 256. [ 17 ] L.A. Lugiato, Phys. Rev. A 33 (1986) 4079; B.J. Dalton, in: Quantum optics IV, eds. J.D. Harvey and D.F. Walls (Springer-Verlag, Berlin 1986) p. 99. [ 18 ] M.O. Scully, K. Wodkiewicz, M.S. Zubairy, J. Bergon, Ning Lu, and J. Meyer ter Vehn, Phys. Rev. Lett., in press.