Collective resonance fluorescence in a strongly squeezed vacuum

Volume 77, number 1

OPTICS COMMUNICATIONS

l,lune 1990

COLLECTIVE RESONANCE FLUORESCENCE IN A STRONGLY SQUEEZED VACUUM J.I. C I R A C and L.L. S A N C H E Z - S O T O

Departamento de ¢Optica.Facultadde CienciasFisicas, UniversidadComplutense. 28040 Madrid. Npain Received 18 December 1989

We study the resonance of a system of two-level atoms driven by an intense pump field and damped by a broadband squeezcd vacuum. The steady-state spectrum is shown to be dependent on the phase between the pumping field and the squeezed vacuum field in strong-squeezing limit. As a consequence, one sideband of the fluorescence triplet may even disappear.

The p r o b l e m o f resonance fluorescence from a two-level a t o m driven by a coherent field and d a m p e d by an electromagnetic reservoir has been the subject o f n u m e r o u s studies [ 1 ]. As it is well known, when the atomic transition is saturated, the fluorescence spectrum shows the three-peaked structure first predicted by Mollow [2] and verified by a n u m b e r o f experiments [3]. In the traditional t r e a t m e n t o f spontaneous emission the reservoir is supposed to be in the vacuum state. However, with the recent successful generation o f squeezed light [4], interest turns to the modifications o f the spectroscopic properties o f the a t o m by the squeezing. In particular, when the reservoir exhibits b r o a d b a n d squeezed fluctuations, the relaxation o f one o f the atomic observables is inhibited and the corresponding linewidth narrowed [5], which leads to i m p o r t a n t modifications in the fluorescence and absorption spectra [ 6 8 ]. These effects are essentially related to the phase-sensitive white noise o f the squeezed reservoir [9]. In this letter we report some exact results on the collective behavior of an atomic system interacting with a squeezed bath and driven by a coherent field. We show that, in the strong-squeezing limit, if the driving field is out o f phase with the m a x i m a l l y squeezed q u a d r a t u r e o f the vacuum, there is a set o f steady-state solutions for which the fluorescence spectrum d e p e n d s on the initial state o f the atomic system. As a consequence, one o f the sidebands o f the fluorescence triplet can disappear, which could be used for detecting a squeezed field through its interaction with a two-level system. The h a m i l t o n i a n describing the interaction o f p two-level a t o m s with the quantized m u l t i m o d e radiation field and a classical driving field is given in the electric-dipole and rotating-wave a p p r o x i m a t i o n by

tt=ho,~oSr +hg2{exp[--i(¢oLt--OL) ]S+ +exp[i(oOLt--~)L) ] S - } + Hrad +h(S+ F+ S - F t) ,

(1)

where ~Oo is the atomic resonance frequency, O)L and ~L are the frequency and initial phase o f the classical driving field; H~ad is the free h a m i l t o n i a n o f the quantized radiation field, F and f'* are bath operators defined in terms o f the positive- and negative-frequency c o m p o n e n t s o f the field, respectively, 2 0 is the Rabi frequency associated with the driving field, and S-~, Sz collective atomic operators satisfying the well-known su (2) algebra. We assume that the q u a n t i z e d r a d i a t i o n field is in a b r o a d b a n d squeezed v a c u u m state centered about frequency (~Opassociated with the field p u m p i n g the squeezed bath. The b a n d w i d t h o f the squeezing is supposed to be sufficiently b r o a d ( m u c h larger than the natural linewidth) that the squeezed vacuum appears as ~-correlated squeezed white noise to the atoms. Then the correlation functions for the bath operators are written in the form [ 10 ]

(Ft(t)F(t')>=TNS(t-t'), 26

(F(t) Ft(t')>=y(N+l)~(t-t'), 0030-4018/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)

Volume 77, number 1

OPTICS COMMUNICATIONS

(F(t) F(t')) =yMexp(-2impt) ~(t-t'),

(Ft(t) F(t'))

=yM* exp(2impt) ~ ( t - t ' ) .

1 June 1990 (2)

Here ~ is the Einstein decay coefficient for the unsqueezed vacuum, and N and M are parameters that describe the squeezing. They must obey the relationship IMI2<~N(N+I),

(3)

with the equality holding for a minimum uncertainty state (MUS), as might be obtained at the output of an ideal parametric oscillator. We shall write M = IMI exp (i~p), and the phase ¢p will depend on the specific details of the scheme used to generate the squeezed vacuum, although for the sake of generality we shall leave it arbitrary. On introducing the transformation fi=RpR*,

(4)

with R =exp [i(copt- Cp/2)Sz],

(5)

the dynamical behavior of the atomic system is described by the master equation obtained by Gardiner and Collet [10], that can be recasted in the following form: 0/~/Ot= - i 3 [Sz,/~] - iO[exp [ - i ( & - 0) ]S + + e x p [ i ( S t - 0) ] S - , p] + (y/2) [ (N+ 1 )L - + + N L ÷ - - IMIL ++ - I M I L - - ]P,

(6)

where/~ is the transformed reduced density operator, A=Wo-Ogp,

~=mL--mp,

0=0L--~p/2,

(7a)

and L '~Pp= 2S'~pS p - SPS'~p-pSPS '~ .

( 7b )

We concentrate in what follows on fields with a large degree of squeezing, i.e., N>> 1. How well this can be realized in practice is a matter of current investigation. An ideal degenerate parametric oscillator operating at 90 per cent of threshold has N ~ 90. In this case N + l ~ ~ l )

(l+l/2N+...),

N~x/~N+l)

(1-1/2N+...),

(8)

and we can expand (6) in terms of l / N retaining only the first term. Using the x and y components of the spin operator, we get 0fi/0t = - i A [ Sz, fi] - 2i~[cos( 6t-O)Sx + sin( & - O ) S y , fi] + (~/2)[~

l)(LXX+LYY)fi+ IM] (LYY-LXX)fi] •

(9)

From the commutation relations it is easily shown that Tr(L'*~Sz)=-, = 0,

ifa¢fl; otherwise,

Tr (S,~ [Sp, fi] ) = T r ( [S,~, Sz]fi) =i~,~& ,

(10a) (10b)

where e~& is the antisymmetric Levi-Civitta tensor and ~r=RtSrR. Using these relations, we easily get the following equation 27

Volume 77, number l

OPTICS COMMUNICATIONS

1 June 1990

(11)

\
\/

where -A

D = ( - ~7(',/N(NT1) + IM] )

- ½ y ( , , / ~ x ~ 1) - I M I )

-2£2 sin(&-O)

2£2 sin ( d t - O) \ -2£2 c o s ( g t - 0 ) ] .

2 0 c o s ( & - 0)

When the squeezed vacuum is squeezed about the frequency of the driving field (~ = 0), these equations have steady-state solutions identical with the ones obtained by taking the limit N ~ o o in the solutions given by Carmichael et al. [6]. However, when the broadband squeezed field is a MUS, the atomic detuning is zero (A = 0) and the driving field is out of phase with the maximally squeezed quadrature of the vacuum (0= n/2), there is an infinite set of steady-state solutions characterized by

For this case, the master equation, in the lowest order in

l/N,

simplifies to

8~/Ot= 2i£2[S,.,/~] + 7NL "V~,

( 13 )

which admits the exact solution fi,.,., ( t ) - (Y[PIY' > =/~yy,(0) e x p [ - T N ( y - y ' )2t+ 2i~2(y-y' ) t ] ,

(14)

where lY> are eigenstates of Sy. All the off-diagonal matrix elements of the reduced density operator in this basis vanish quickly with time, a fact previously noticed by Carmichael et at. [6], and that may be used to prepare atoms in certain coherent superpositions of Dicke states [ 11 ]. We wish to point out that eq. (14) does not give the exact steady-state regime if we consider all the orders in (8), but for a time scale [½7(N- [M[ + 1/2 ) ] - ~>> t>> (),N) -J it can be interpreted as a quasi-stationary solution. It is precisely in this time scale where our results are valid. Note also that in the case in which N is infinite, although the expansion (8) holds, the master equation (9) is not valid because the width/1 of the band over which the vacuum is squeezed must be larger than the largest decay constant of the atom [12], i.e.,/~ >> 7N. The fluorescence spectrum for the system is calculated as the Fourier transform of the autocorrelation function of the atomic dipole operator ( S + (0) S - ( r ) ) in the stationary state and using the quantum regression theorem. The final result is < ~ ( 0 ) > - < ~ ( 0 ) > + 7XS~+S - + (7N)2+ (¢O--OJL+2£2) 2 4~ (yN)2+ (~--~OL--2£2) 2

yNS~+SS(¢o)= fi(OJ--O~L)+ 47~

(1 5)

where S=p/2. This scheme for calculating the spectrum assumes that there should be a window of unsqueezed vacuum modes through which we can view the fluorescence [6]. However, when there is a large number of atoms (p>~N>> 1) the intensity of the spectral lines are of the order o f p 2, will dominate over the squeezed noise and, hence, such a window is unnecessary [7]. The spectrum is a triplet peak for £2#0. The central peak has a height proportional to the mean value of the square of the projection of the Bloch vector on the direction of the low-noise quadrature phase, and has zero width since we have considered that S~, does not evolve with time. In fact, the central peak will have a linewidth which is negligible in comparison with the widths of the sidebands in the strong squeezing limit. The new and interesting feature of the spectrum is the dependence of the height of the sidebands on the initial state. When initially all the atoms are inverted the two sidebands are identical. But when the average of ~, is not zero, the sidebands are different. In fact, one of then can disappear when the initial state of the system is 28

Volume 77, number 1

OPTICS COMMUNICATIONS

1 June 1990

in an eigenstate of Sy with eigenvalue p/2. For the sake of simplicity we study the case of a single two-level atom initially prepared as a coherent superposition of the ground ( I g ) ) and excited ( l e ) ) levels, I ~v) = c o s ( ~ / 2 ) I g ) + sin ( ~ / 2 ) e x p ( i ~ , ) l e ) •

(16)

In such a case, the fluorescence spectrum is given by

S(m)=~6(cO_mL)+ TN 1--sin~cos(0L--V) yN I + s i n ~ c o s ( 0 L - - ~ ) 87Z (yN)2+(m--OJL+2Q) 2 + 8X (7N)2+ (CO--COL--2Q) 2"

(17)

In fig. 1 we have plotted the spectrum for different angles. When the initial atomic dipole is in phase with the driving field, and ~=zc/2, one of the sidebands disappears and the other, due to energy conservation, is enlarged. A similar result has been recently reported by Courty and Reynaud using a semiclassical dressedstate approach. In their analysis the disappearance of the sideband is a manifestation of the dressed population trapping due to the existence of a detuning between the atomic and the pump frequency [ 13 ]. In a many-atom system, the same kind of spectra are found, but with more intense lines. Finally, it is interesting to compare the analytical spectrum in ( 17 ) with the exact transient spectrum measured in the time interval where the approximation of N large is valid, using an exact numerical solution of the complete master equation and the quantum regression theorem. This physical transient spectrum is given by [14] to+ T

to+T

dh J d t 2 e x p [ - ( A - i m ) ( t o + T - h ) ] exp[-(A+im)(to+T-t2)l(S+(tt)S-(t2))

s(o~) = to

.

to

(18) Here 1/A is the filter's spectral transmission function and (to, to+ T) is the time interval at which the measurement takes place, i.e. (½7 ( N - IM] + 1/2 ) ) - t >> to, to + T>> (7N)-1. We have chosen the normalization in such a manner that coincides with the formula used for the stationary state when the measurement time T tends to infinite and A remains small. In fig. 2 we have compared the analytical spectrum ( 17 ) with ( 18 ) for 0L = ~' and ~= ~/2. For N = 100 (fig. 2a) and larger, both results are practically identical (note that we have not plotted the delta function appearing in ( 17 ) ), according with the theoretical results. For N-- 5 (fig. 2b) we observe that the plots differ in minor details, but the qualitative behaviour is identical. In conclusion, when a system of two-level atoms is driven by a resonant coherent field and damped by a squeezed vacuum, and the driving field and the squeezing are out of phase, a quasi-stationary state is reached in the time interval in which the fast decaying polarization quadrature of the system is in its stationary state and the other polarization quadrature has been not affected yet. This time interval is enlarged considerably in the strong squeezed limit. The existence of this state can be viewed in the fluorescence spectrum: depending on the initial state of the atom one of the sidebands of the spectrum may disappear. This fact can be used for detecting squeezing through the interaction with a two-level system.

60

a./2

Fig. 1. Fluorescence spectrum of the light emitted from a twolevel atom, initially in the state (16) with ~= n/2, damped by a strong squeezed vacuum and driven by a coherent field, as a function of the phase difference between the external field and the atomic dipole moment. Only the sidebands have been plotted. Note that for ~L--~/]~-0, 7[one of the sidebands disappears. 29

Volume 77, number

OPTICS COMMUNICATIONS

O.05

a

II.Ot

0.04

0.03

0.0:]

N'~S(<~)

b

N.ys(~)

t / /

().02

1 June 1990

0.02

i / / / /'

II.lJl

0.oo

.

.

.

0.01

.

.

.

.

.

.

.

.

.

.

.

0.00

-

-'3

o --

N~

to L

N"t

Fig. 2. Comparison between the stationary fluorescence spectrum resulting from ( 17 ) (dashed line) and the transient spectrum defined in (18) (solid line) with ~=~/2, 0L=~u, g2/7=N,A/)'= 1, 7to=5/N. )'T=N, and N = 100 (a) and N = 5 (b).

We are much i n d e b t e d to Prof. E. Bernabeu for his continual advice and interest in the present work.

References [ 1 ] For a review see J.D. Cresser, J. H~ger, G. Leuchs, M. Rateike and H. Walther, in: Dissipative Systems in Quantum Optics, ed. R. Bonifacio (Springer, Berlin, 1982) p. 21. [2] B.R. Mollow, Phys. Rev. 188 (1969) 1969. [3] F. Schuda, C.R. Stroud and M. Hercher, J. Phys. B 7 (1974) L198~ F.Y. Wu, R.E. Grove and S. Ezekiel, Phys. Rev. Lett. 35 (1975) 1426; W. Harting, W. Rasmussen, R. Schieder and H. Walther, Z. Phys. A 278 (1976) 205. [ 4 ] For a review see the special issues on squeezed states: J. Mod. Opt. 34 ( 1987 ) 709; J. Opt. Soc. Am. B 4 ( 1987 ) 1449. [ 5 ] C.W. Gardiner, Phys. Rev. Lett. 56 ( 1986 ) 1917. [ 6 ] H.J. Carmichael, A.S. Lane and D.F. Walls, J. Mod. Opt. 34 (1987) 82 I. [7] T. Quang, M. Kozierowski and L.H. Lan, Phys. Rev. A 39 (1989) 644. [8l A.S. Parkins and C.W. Gardiner, Phys. Rev. A 37 (1988) 3867; H. Ritsch and P. Zoller, Phys. Rev. A 38 ( 1988 ) 67; Optics Comm. 64 ( 1987 ) 523; 66 ( 1988 ) 333: S. An, M. Sargent and D.F. Walls, Optics Comm. 67 (1988) 373. [ 9 ] G.M. Palma and P.L. Knight, Phys. Rev. A 39 (1989) 1962; A.K. Ekert, G.M. Palma, S.M. Barnett and P.L. Knight, Phys. Rev. A 39 (1989) 6026; G.M. Palma and P,L Knight, Optics Comm. 73 (1989) 131; K. Zaheer and M.S. Zubairy, Phys. Rev. A 39 (1989) 2000. [ 10] C.W. Gardiner and M.J. Collet, Phys. Rev. A 31 ( 1985 ) 3761. [ 11 ] G.S. Agarwal and R.R. Puri, Optics Comm. 69 (1989) 267. [ 12] C.W. Gardiner, A.S. Parkins and M.J. Collet, J. Opt. Soc. Am. B 4 (1987) 1683. [ 13 ] J.M. Courty and S. Reynaud, Europhys. Lett. 10 ( 1989 ) 237. [ 14] J.H. Eberly and K. W6dkiewicz, J. Opt. Soc. Am. 67 (1977) 1252.

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