Radiative properties of strongly driven atoms in the presence of ...

340

J. Opt. Soc. Am. B/Vol. 10, No. 2/February

1993

T. W Mossberg and M. Lewenstein

Radiative properties of strongly driven atoms in the presence of photonic bands and gaps T. W Mossberg and M. Lewenstein* Department of Physics, University of Oregon, Eugene, Oregon 97403 Received April 30, 1992; revised manuscript received July 30, 1992

We show that the presence of photonic bands and gaps dramatically modifies the properties of single-atom resonance fluorescence. When, for example, photonic gaps suppress radiative emission on one or both of the strong-field resonance fluorescence sidebands, a driven atom is found to emit monochromatic radiation that is insensitive to fluctuations in the driving-field intensity and does not display the photon antibunching that is normally associated with single-atom resonance fluorescence. In such situations the atom's radiative properties come to resemble those of a perfectly classical dipole.

1.

INTRODUCTION

Spontaneous decay of an atomic transition is usually assumed to be characterized by a single excitation-field-

independent rate, the Einstein coefficient. However, it was suggested' and clearly demonstrated 2 that

spontaneous-emission rates may be substantially modified by environmental factors that lead to changes in the coupling of the emitting atom to the electromagnetic vacuum. These changes lead not only to modifications of spontaneous-emission rates but also to other interesting effects, such as cavity-modified Lamb shifts,3 non-Markovian decay,4 and velocity-dependent spontaneous emission.' Changes in spontaneous-emission rates have frequently been attributed to changes in the local density of electromagnetic modes, and, early on at least, mode-density changes were attributed to the effects of cavities, waveguides, and so on. Yablonovich, in 1987, suggested that

the mode-density variations would arise in the form of photonic band structure within periodic dielectric

structures.6 '7 Spontaneous radiative processes become particularly interesting when atoms interact with an electromagnetic mode density that has been perturbed in a frequencydependent manner (i.e., is colored). In a recent letter we demonstrated, for example, that the mechanical effects of light take on a dramatically different character in the presence of a colored vacuum. We found that lightinduced forces and the atomic velocity ranges over which they are effective can, in colored vacuums, be much larger than the corresponding free-space quantities. More relevant to the present considerations, it was recently pointed out that colored reservoirs give rise to spontaneous radiative processes that are dynamically controllable. Examples of such effects are the dynamic suppression of spontaneous emission 9 and vacuum-field dressed-state pumping.'

0

In the present paper we propose to study the dynamic modification of spontaneous emission in the presence of photonic band structure, concentrating specifically on the problem of resonance fluorescence from strongly driven atoms." The effects that we discuss depend critically on the fact that the electromagnetic mode density is strongly 0740-3224/93/020340-06$05.00

colored by the presence of photonic bands. Throughout this paper we assume that the photonic mode density, while globally colored, is constant over the immediate spectral regions surrounding each characteristic atomic resonance. Concomitantly we assume that the response of the photon reservoir is Markovian and immediate. Situations involving non-Markovian reservoir response were treated by us elsewhere.4 9 The photonic mode density is taken as zero within photonic band gaps, but this extreme is not required for the appearance of effects that are qualitatively similar to those predicted here. We analyze several generic situations in which transitions at one or more of the three frequencies of the strongfield resonance fluorescence triplet are suppressed owing to the presence of photon gaps. The paper is organized as follows. We first introduce our model and describe it in terms of dressed states 2 in Section 2. In Section 3 we incorporate the photon bands and band gaps into the model and discuss the method of its solution. Finally, in Section 4 we present our results, which are as follows: * If one or two strong-field resonance fluorescence sidebands are suppressed, the atom behaves like a perfect classical dipole and is the source of coherent monochromatic radiation that is insensitive to fluctuations of the driving-field intensity. * If one strong-field resonance fluorescence sideband and the central frequency are suppressed, the atom does not produce any fluorescence light at all. * Finally, if the central frequency is suppressed, the spectrum consists of two inelastic sidebands only and does not contain an elastic Heitler peak. In all cases photon antibunching, a fundamental aspect of atomic fluorescence in free space, is not displayed (for a recent review of photon counting see Ref. 13 and references therein).

2. STRONGLY DRIVEN ATOM: DRESSED-STATE PICTURE We consider a two-level atom (ultimately located in a medium with photon gaps and bands but initially assumed to ©91993 Optical Society of America

Vol. 10, No. 2/February


be in free space) that is pumped by an external driving field of frequency L. The strength of the pump is characterized by the Rabi frequency fl. We denote the bare atomic transition frequency by o, so that A, = o - OL denotes the atom-laser detuning. Let U3, S, and -adenote the standard Pauli matrices that describe a two-level atom in the bare atomic-state basis. The density matrix of the system obeys the Liouvillevon Neumann (master) equation of the form'4 p The Hamiltonian

=

-i(e, p) + sEAP.

(1)

lower-sideband transitions from the state. Accordingly, we write

where the rotating-wave approximation is employed to eliminate the explicit time dependence of the pump field at the frequency OL. The last term in Eq. (1) describes spontaneous emission, i.e.,

(7)

Here p is also evaluated in the dressed-state basis. The above decomposition turns out to be quite useful, since it allows one to identify the transition frequencies that correspond to the specific parts of the Liouvillian. The central term, ST, for instance, describes transitions between the same dressed states, i.e., transitions at the central frequency OL, and Lep = (/2)sin 2 (2a)(0-3p3

(2)

St)] '

I-) to the +) dressed

-TAP= S4p + SU.P+ SIp-

in Eq. (1) is given by the expression

+ l( C + Xe 1/2 [Al1(}3

341

1993/J. Opt. Soc. Am. B

-

p).

(8)

The u term describes transitions at the upper-frequency fluorescence sideband, i.e., at the frequency OL + f', and SeuP =

2 [1 + cos(2a)]2(o

po-t

-

I

cirp

1

t (9)

-TAP= 2 y(Upat - 1/2

1/2pa

),

(3)

where 2 y denotes the free-space atomic spontaneousemission rate (equal to the Einstein's coefficient A). Expression (3) is modified in the presence of photonic bands

Finally, the term describes transitions at the lowerfrequency fluorescence sideband, i.e., at the frequency

(WL - 9',and

and gaps, as we show in Section 3.

Equations (1)-(3) are written in the basis of the atomic Hilbert space, consisting of the bare excited state 11)and bare ground state I0). To obtain a dressed-state picture, we have to change the basis, introducing dressed states

1+)= (cos a)11) + (sin a)10), |-) = -(sin

a)|1) + (cos a)10).

(4) (5)

In the above expressions the rotation angle a, which belongs to the interval [0,s/2], is defined through the rela-

tions fl = ff sin(2a) and A, = f'cos(2a), where f'

denotes the effective Rabi frequency and is equal to the 2 2 2 dressed-state energy splitting ff = (V + A1 )1/ . We dressed (lower) refer to the l+) (I-)) state as the upper state, since it is the higher- (lower-) energy eigenstate. The details of the transformation to the dressed-state basis can be found, for instance, in Ref. 15. After we perform the unitary transform defined by relations (4) and (5), the Hamiltonian,

Eq. (2), becomes Wet= (Q/2)U3,

(6)

where a 3 , Art, and u now refer to the atomic operators in the dressed-state basis, corresponding to the dressed-state inversion, raising, and lowering operators, respectively. Note also that the atomic Hamiltonian, Eq. (6), describes oscillations of the dressed-state polarization with the frequency ff. These oscillations are fast when Q' is large. The transformation to the dressed-state basis is also applied to the spontaneous-emission term, Eq. (3). In the limit of very strong driving (when 1' >> y) it is sufficient to keep only the resonant terms, i.e., those that change on a time scale slower than 1/fY (see Refs. 12 and 15). The

spontaneous-emission term then becomes the sum of three terms that describe (i) spontaneous transitions between the same dressed states, (ii) upper-sideband transitions from the l+) to the I-) dressed state, and (iii)

2Lip= 2[1- cos(2a)]2(tp

10-

o-pt (10)

The terms in parentheses that include 0r3pT3, CPO' t, and oatpo in Eqs. (8)-(10), respectively, describe spontaneous emission on the corresponding transition. The coefficients of these terms describe spontaneous-emission rates. This identification will prove important when we turn to the calculation of photon statistics.16 PHOTONIC BANDS AND GAPS Since we have identified the transition frequencies that correspond to the specific terms in the Liouvillian' and corresponding rates, it is simple to incorporate a frequencydependent photonic vacuum into the model. We do this now so that we may consider the case of photonic bands and gaps in Section 4. Let us assume that the photonic density of modes is frequency dependent and changes on a frequency scale of the order of fY. On the other hand, assume that it is practically constant when viewed on frequency scales larger than y in the spectral vicinity of the central and the sideband frequencies. Since the density of modes does not change rapidly in the vicinity of the dressed-state transition frequencies, its only effect is differentially to modify the three spontaneous dressed-state emission rates. Let us denote factors that modify spontaneous-emission rates at the central, upper-sideband, and lower-sideband transition byp, p,, and pi, respectively. The density matrix can then be written as 3.

p = STp= -i(W',p) + PcScP + PuSuP + P1SLIP- (11) For convenience we introduce the following expressions for the parts of the Liouvillian that describe the emission of central, upper-, and lower-sideband photons,

342


1993


Sp = P(y/2)sin2 (2a)- 3pcr3 , SIuP= p(y/2)[1 +

tpa ,

p = ip + 0Jcp+ SIuP+ SIP,

(14)

so that the Liouvillian that governs evolution between emissions is defined as sl = - 0-1 The master Eq. (11) leads to the following evolution equations for matrix elements. The population of the upper dressed state evolves in time in accordance with

-

- cos(2a)]2 p__ 2)[1 2

(2c)p++ + pl(y/2)[l

p(y/2)sin2

(2a)p++ - p(y/

- p(y/2)[

- cos(2a)] 2p_. (16)

In Eqs. (15) and (16) the first two terms on the right-hand sides that come with positive signs originate from the emission processes. Other terms describe evolution between successive emission acts and follow from the action of the Liouville operator . The off-diagonal elements of the density matrix evolve according to =

2

-

2 + - cos(2a)]

p[1 + cos(2a)]2 2 pu[l + cos(2a)] (20)

The stationary dressed-state polarization is proportional to the off-diagonal matrix elements, and these vanish in

the stationary limit [i.e., lim,-.p+-(t) = lim,-.p_+(t) = 0]. The relaxation rates of the dressed-state polarization and inversion are, respectively, 1 = YP, sin 2 (2a) + (/4)pU[1 + cos(2a)]'

+ (/4)PJl 72

-

cos(2a)]2,

(21)

2

= (7/ )p.[l + cos(2a)]2 + (y/2)pu[l

-pc(y/2)sin 2 (2a)p+ - i+ -

p(y/4)[1 + cos(2a)]2 p+-

-

pi(y/4)[1 - cos(2a)] 2 p+-.

(17)

In Eq. (17) only the first term on the right-hand side originates from the emission processes. The equation for p+ is obtained by taking the complex conjugate of Eq. (17). It is elementary to find solutions of Eqs. (15)-(17). We do not present the full expressions, since they are somewhat lengthy, and it turns out that simplified forms are suitable for the parameter range of interest here. Before concluding this section, we introduce the following additional definitions and results. Wedenote the stationary occupation of the upper dressed state by P+. It is easy to con-

firm that

(22)

The above general results allow us to obtain specific results that correspond to various arrangements of photonic bands and gaps.

We now turn to the discussion of fluorescence emitted by a strongly driven atom that is located in a medium in which photonic bands and gaps are present. We consider various possible situations in which one or two of the three strong-field resonance fluorescence frequencies are located within a gap. Spontaneous transitions on the gap-frequency transitions then will not occur. For simplicity we limit our considerations to the case of perfect bare-atom-driving-field resonance, i.e., l = 0. In this case sin(2a) = 1, while cos(2a) = 0. Below we present results for the following quantities that characterize the atomic fluorescence:

S(co)= lim Re

f

exp[-i(c -

L)r]i'V(t

2

-

PDl - cos(2a)] cos(2a)]2 + PU[l + cos(2a)]2

Similarly, stationary occupation of the lower dressed state is =

Pu[1 + cos(2a)]

pl[-

cos(2a)]2

2

+ pU[1+ cos(2a)]2

(19)

(

Obviously P_ = 1 - P+. The stationary inversion is thus

+

rT)Y+'(t)dT, (23)

where c6`' denote positive and negative frequency compo-

nents of the scattered electric field. In the case of Markovian, but frequency-dependent, reservoirs t(+) are given by %(7(t) = [(Y/ 2 )pc sin 2 (2 a)] 1/23(t)

+ {(y/2)pJl + cos(2a)]2}/ 2u(t) -

{(/2)pl[1 - cos(2a)]2}"12at (t),

1_(-'(t) = [ (+'(t)]t.

pl[l

cos(2a)]2 .

1. The stationary power spectrum of resonance fluorescence S(Cw) is defined as

- p(y/2)sin2 (2a)p+-

P=

-

4. RESONANCE FLUORESCENCE IN THE PRESENCE OF PHOTONIC BANDS AND GAPS

p^(y/2)sin2(2a)p- + p(y/ 2)[1 + cos(2a)] 2p++ - p(y/2)sin 2(2a)p-

pi[

cos(2a)]

+ cos(2a)] p++.

The population of the lower dressed state fulfills =

-

(13)

We may now split the full Liouvillian 2 into an emission part and a part that describes the evolution of the system between successive photon-emission acts.' 6 We may write

P++=p(y/2)sin

pi[

(12)

9iip=pi(y/2)[1 - cos(2a)]2c t pr.

2

p-

t

cos(2a)] 2

(24) (25)

2. The two-time correlation function wx(T) is defined as the conditional probability density of detecting the next x-type photon at time t + provided that a photon of y type was detected at time t. Here x andy may denote c, u, and 1. The function w(T) is defined as 6 WXY(r)= lim Tr[9J. exp(s4ir)yp(t)]

t

where p(t) = exp(2t)p(0).

Tr[ffp(t)]

(26) (6

The function wxy(T)describes



precisely the statistical properties of the fluorescence radiation. For example, its short-time behavior determines whether photons are bunched or antibunched (see Refs. 13 and 17 and references therein). 3. The two-time correlation function gxy(T) may be regarded as the conditional probability density of detecting an x-type photon at time t + , provided that a y-type photon was detected at time t. As above, x and y may denote 7 c, u, or 1. The function gy(T) is defined as' gXy(T)

=

exp(T)Syp(t)] lim Tr[J- Tr[9Typ(t)] t-11

One fluorescencesideband within a photonic band gap: For definiteness, let us assume that the frequency of the

upper-sidebandtransition at frequency (OL + i' lies within a photon gap, where the density of photon modes is zero, i.e., pu = 0. Let us also assume that the density of photon modes has the same value at the two remaining strong-field resonance fluorescence frequencies. In particular, we take Pc = pi = 1. Suppression of emission at the upper sideband leads to a complete polarization of the population within the dressed states. One finds that in steady state P+= 1,

(28)

P_ = 0.

(29)

The atomic-emission spectrum that is characteristic of this steady state consists of a single peak at the frequency of the central fluorescence component, i.e., at (WL. Before the steady state is achieved, at most one l-type photon can be emitted. The power spectrum of the steady-state fluorescence consists of only the purely elastic Heitler delta peak, i.e., CWL),

(30)

which indicates that the atom is acting like a perfect linear scatterer, such as a harmonic oscillator. 8 The scattered light displays an exponential two-time correlation function, i.e., wcc(t) = (y/2)exp(-yT/2),

343

It is interesting to expand on the comparison of the band-gap-modified atomic radiator with the harmonic oscillator. Harmonic oscillators are linear, and thus so is their response. As a result their scattered field is proportional to the driving field, and the statistical properties of the scattered radiation are fully determined by the statistical properties of the pump. In particular, any amplitude or phase noise present on the pump is transferred to the scattered signal. Note that the atomic dipole attains its maximal possible value in the steady state of the sys-

tem under consideration and that the steady state is .(27)

Obviously, for short times the functions wy(,) and g.y(T) are identical. Having defined the quantities of interest, we now evaluate them for four different photon band and band-gap environments:

S(co) = (y/2 )5(w -


(31)

which additionally implies that gcc(&)= y/2 is a constant. A constant g function is characteristic of coherent states and associated Poissonian statistics. It is worth stressing that our strongly driven atom behaves quite diferently from a normal free-space atom. Its spectrum is compressed from three nonzero-width peaks to a single delta-width peak, and its photons lose their classic antibunched character. The atom behaves in the present case as a perfect classical scatterer. It scatters photons only elastically, as harmonic oscillators do. From the point of view of the bare-state picture, what happens is that an atomic dipole builds up and becomes a source of coherent radiation.

insensitive to changes in the driving-field amplitude, provided that the fluorescence sideband within the photonic gap remains in the gap. This implies that our driven atom is, in contrast to the harmonic oscillator, insensitive to amplitude fluctuation of the pump and will give the same output over a wide range of driving-field strengths. This suggests various applications of this sytem, as we discuss below.

Both resonancefluorescencesidebands within photonic band gaps: Here we assume that the frequencies of both the upper- and the lower-sideband transitions WL ± f' lie within a photonic gap, where the density of photon modes is zero, i.e., p11 = pi = 0. We also assume that the density of photon modes at the remaining frequency is Pc = 1. In this case the atomic evolution is not ergodic, and the final state depends explicitly on the initial state. The offdiagonal elements of the density matrix vanish in the stationary state, but the initial occupations of the upper and the lower dressed states are conserved. The atom thus ends up in a mixed state, characterized by P+ = P+(O)I

(32)

P = P ().

(33)

As in the previously discussed case, however, the atom emits photons only at the central frequency. In fact, it again acts like a perfect linear scatterer, such as a harmonic oscillator, and scatters pump photons purely elastically. The power spectrum of the fluorescence radiation again consists only of the Heitler delta peak, Eq. (30). In the stationary state the emitted photons are thus of the c type. In the present situation none of the u- or i-type photons may be emitted even in the transient regime. There are two independent sequences of c-type photons emitted in a purely coherent way. One comes from the transitions between the upper dressed states, and the other comes from the transition between lower dressed states. The photon statistics are Poissonian,' 8 since all photons are emitted independently one from another and at the same rate. The probability distribution for the temporal spacing between emissions is thus again expo-

nential, Eq. (31). Equation (31) again implies that gcc(T)= y/2 is constant, as is characteristic for coherent states and Poissonian statistics. We stress again that our strongly driven atom behaves in this case as a perfect classical scatterer, and the relevant discussion above applies to the present case as well.

Central fluorescence component within a photonic bandgap: In this case the atom cannot radiate at the central frequency, and thus scattering must be purely inelastic. Here we assume that the frequencies of both the upper- and the lower-sideband transitions WOL ± if lie in

344


1993


frequency ranges where the density of photon modes is nonzero, i.e.,p,. = p = 1. On the other hand, we assume that density of photon modes at the central frequency vanishes, Pc = 0. In the stationary state the density matrix is diagonal,

and the upper and the lower dressed state levels are equally populated, i.e.,

P+= 1/2, P = 1/2.

(34)

(35)

The spectrum of the fluorescence radiation consists of two inelastic peaks centered at WOL f, S(w) =

V

4[(o+

WL+ f) 2 + (7/2)2] 7

4

- C XLL

-

)

2

+ (y/2)2]

(36)

w.1(t) = w(r) = (/2)exp(-yr/2).

(37)

It is possible to define other photon correlation functions, &11(T), VUU(T) that correspond to conditional probability densities of emitting the next photon of some sort at time t + T, provided that a photon of the same sort has been emitted at time t, and a photon of a different sort sometime in the interval [t, t + T]. Such a function can be constructed by convolving Wiu(T) and Wul(T) and is equal to =

11

&(T) = (y 2/4)T

exp(-yT/2).

5.

CONCLUSIONS

We have shown that the resonance fluorescence of atoms

in photonic band-gap materials is strongly modified relative to the case of free-space atoms and has various interesting properties. The effects predicted here are essential to the understanding of optical processes that involve strongly driven atoms in photonic band materials, such as these instances:

* Optical instabilities in such materials will be

There are now two kinds of photons emitted, u- and -type photons. Emission of each of the photons is an independent act, but the frequency sequence of photons is totally correlated: a u-type photon must come after an i-type photon and vice versa. This means that the functions w u(T)and w&(), defined as in Eq. (26), both vanish. The numbers of photons emitted into the u and the I modes are highly correlated.' 9 Note that this relationship is reminiscent of photon-number correlations found in situations that involve nondegenerate parametric downconversion processes and squeezed states.20 Here, however, the two photons are emitted at different times, and the resulting temporal statistics are characteristic of coherent states. In particular, the probability density of temporal spacings between successive emissions is exponential,

tuu(t)

of photon modes at the frequency of the lower sideband transition is p1 = 1. This is a case of a perfect optical radiation trap. The atom stays forever in the upper dressed state either from the beginning or after performing one transition from the lower to the upper dressed state. In the best case the system will emit just one photon. Obviously the notions of a photon spectrum and photon statistics do not apply to this situation.

(38)

This function clearly exhibits antibunching of photons of the same sort. The g functions now have a little more complicated form, since two kinds of photons may and will be emitted. We obtain

strongly modified. For instance, optical bistability will follow quite different input-output relations for output fields that cause a fluorescence sideband to shift into the photon gap region. * Light transmission through such media will be quite different from the usual case, especially when one fluorescence sideband is suppressed. The medium will behave then as if it consists of harmonic oscillators. * Absorption in such media will have a purely quantum character. In the standard case atomic dipoles do have components that are in quadrature with the driving field, and the field produced by a plane of such dipoles may interfere destructively with the driving field. This standard classical picture of absorption does not apply in photonic band-gap material. When one fluorescence sideband is suppressed, atomic dipoles oscillate purely in phase with the driving field. * Wave-mixing processes in photonic band-gap materials may be employed to produce amplitude stabilized output signals. This possibility follows from the insensitivity of atomic emission to pump-power fluctuations whenever the fluctuations do not cause a fluorescence sideband to leave a photonic gap. We leave a detailed discussion of these effects to future publication, announcing here only the possibilities of the new branches of optics associated with photon band-gap materials. *Permanent address, Institute for Theoretical Physics, Polish Academy of Sciences, 02-668 Warsaw, Poland.

REFERENCES gu(T)

= g(7)

gul(T) = g(T)

= =

(/4)[1 - exp(-YT)],

(39)

(/4)[1 + exp(-yr)].

(40)

Central strong-field resonancefluorescencecomponent and one sideband within a photonic gap: Here we assume that the frequencies of the central CUL and, say, 0 upper-sideband transitions OL + nolie within a photonic gap, where the density of photon modes is zero, i.e., Pc = p, = 0. On the other hand, we assume that density

1. E. M. Purcell, Phys. Rev. 69, 681 (1946); D. Kleppner, Phys. Rev. Lett. 47, 233 (1981). 2. W Jhe, A. Anderson, E. A. Hinds, D. Meschede, L. Moi, and S. Haroche, Phys. Rev. Lett. 58, 666 (1987); S. Haroche and D. Kleppner, Phys. Today 42(1), 24 (1989). 3. D. Heinzen and M. S. Feld, Phys. Rev. Lett. 59, 2623 (1987).

4. M. Lewenstein, J. Zakrzewski, T. W Mossberg, and

J. Mostowski, J. Phys. B 21, L9 (1988); M. Lewenstein, J. Zakrzewski, and T. W Mossberg, Phys. Rev. A 38, 808 (1988).

5. Z. Bialynicka-Birula,

P. Meystre, E. Schumacher, and


6. 7.

8. 9. 10. 11. 12.

M. Wilkens, University of Arizona, Tucson, Ariz. 85721 (personal communication, 1991). E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987). E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991); E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Pappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991). T. W Mossberg, M. Lewenstein, and D. J. Gauthier, Phys. Rev. Lett. 67, 1723 (1991). M. Lewenstein, T. W Mossberg, and R. J. Glauber, Phys. Rev. Lett. 59, 775 (1987); M. Lewenstein and T. W Mossberg, Phys. Rev. A 37, 2048 (1988). Y Zhu, A. Lezama, T. M. Mossberg, and M. Lewenstein, Phys. Rev. Lett. 61, 1946 (1988). B. R. Mollow, Phys. Rev. 188, 1969 (1969); see also J. D. Cresser, Phys. Rep. 94, 47 (1983); P. A. Apanasevich, Opt. Spectrosc. (USSR) 16, 387 (1964). and S. Reynaud, J. Phys. B 10, 345 C. Cohen-Tannoudji (1977).

13. J. Grochmalicki and M. Lewenstein, Phys. Rep. 208, 189 (1991).



345

Statistical

Theo-

14. See, for instance, G. S. Agarwal, Quantum

ries of Spontaneous Emission, Vol. 70 of Springer Tracts in Modern Physics (Springer-Verlag, Berlin, 1974).

15. J. Zakrzewski, M. Lewenstein, and T. W Mossberg, Phys. Rev. A 44, 7177 (1991). 16. P. Zoller, M. Matrhe, and D. F Walls, Phys. Rev. A 35, 198

(1987); E. B. Davies, Quantum Theory of Open Systems (Academic, New York, 1976); M. D. Srinivas and E. B. Davies, Opt. Acta 28, 981 (1981); B. R. Mollow, Phys. Rev. A 12, 1919 (1975). 17. H. J. Carmichael and D. F. Walls, J. Phys. B 9, 1199 (1976).

18. R. J. Glauber, in Quantum Optics and Electronics, C. D. A. Blandin and C. Cohen-Tannoudji, eds. (Gordon & Breach, New York, 1965); Phys. Rev. Lett. 10, 84 (1963); Phys. Rev. 131, 2529, 2766 (1963). 19. A. Aspect and G. Roger, Phys. Rev. Lett. 45, 617 (1980). 20. H. J. Kimble and D. R Walls, eds., feature issue on squeezed states of the electromagnetic field, J. Opt. Soc. Am. B 4, 1449 (1987); R. Loudon and P. L. Knight, eds., special issue on

squeezed states, J. Mod. Opt. 34, 709 (1987).