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Photon density-density correlation spectrum of a three-level atom at high photon densities
Physica 105A (1981) 203-218 ~) North-Holland Publishing Co. PHOTON DENSITY-DENSITY CORRELATION
SPECTRUM OF A T H R E E - L E V E L ATOM AT HIGH PHOTON DENSITIES* Constantine MAVROYANNIS Division o[ Chemistry, National Research Council of Canada, Ottawa, Ontario, Canada K 1A OR6 Received 2 July 1980
The excitation spectrum arising from photon density--density correlations has been calculated for two types of a single three-level atom for which electronic transitions occur: from two different upper levels to a common lower one and from a common upper level to two different lower levels, respectively. Photon density-density correlations in such a system are described by the interference spectrum, which results from the beating between the two electronic transition frequencies. The spectral function has been calculated in the limit of high photon densities and consists of three pairs of bands, which are similar to those of the interference spectrum when the main peak at the frequency to = A is missing, where A is equal to the frequency splitting between the excited states of the atom. The probability amplitude for the occurrence of each pair of bands depends on the parameter 7/= A2/02, where/] is the Rabi frequency. Numerical values indicate that the probability amplitudes become negative, zero, or positive for different values of the parameter 7. Using the spectral function, an expression has has been derived for the photon density-density correlation function, which describes photon density-density correlations at two different times and at finite temperatures. In the limit of equal times, the mean square photon density distribution function has been calculated and discussed. At zero temperatures, numerical values for the mean square photon density distribution function indicate that it is positive for "11<0.5, it vanishes for values of rt approximately 0.7> "0 >0.5, and it becomes negative for "0 - 0.75. The possibility of observing photon density-density correlations is discussed.
1. Introduction T h e r e has b e e n c o n s i d e r a b l e i n t e r e s t in the f l u c t u a t i o n s p e c t r a of laser beams~-5). M o s t of the i n v e s t i g a t i o n s h a v e b e e n d e v o t e d to s t u d y i n g the effects of c o o p e r a t i v e a n d a t o m i c i n t e r a c t i o n s o n the p t h o t o n statistics of a single m o d e laser. E x p r e s s i o n s for the p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n of the p h o t o n o c c u p a t i o n n u m b e r as well as e x p e c t a t i o n v a l u e s of the light i n t e n s i t y a n d i n t e n s i t y c o r r e l a t i o n s h a v e b e e n d e r i v e d b y m a k i n g use of different m a t h e m a t i c a l t r e a t m e n t s ; we r e f e r to a r e c e n t p a p e r b y Zubairy5), w h e r e details a n d r e f e r e n c e s c a n b e f o u n d . T h e p h o t o n c o r r e l a t i o n or p h o t o n b u n c h i n g p h e n o m e n o n has b e e n first d e m o n s t r a t e d b y H a n b u r y B r o w n a n d T w i s s 6) a n d d e s c r i b e s the d i s t i n c t t e n d e n c y o f p h o t o n s in a light b e a m e m i t t e d b y a t h e r m a l e q u i l i b r i u m s o u r c e * Issued as NRCC No. 18708. 203
204
CONSTANTINE MAVROYANNIS
to arrive at a detector in bunches or correlated pairs. The bunching effect has been studied implicitly in time-resolved correlation experiments, where photon field correlations have been measured with the help of photoelectric detectors. Glauber 7) has shown that the joint probability density of photodetection is proportional to the normally ordered correlation function, which is of the fourth order in the field operators, or of the second order in the light intensity; we refer to the review paper by Mandel and Wolf a) where a detailed discussion and literature are given. The photon antibunching effect or the anticorrelation effect in the resonance fluorescence from a single two-level atom has been discussed by Stolerg), Carmichaei and Wallsl°), Kemble and Mandelll'12), Kozierowski and Tanfisl3), Cohen-Tannoudji and Reynaud ~4) and was observed first by Kimble et al. ~5) and later by Degenais and Mandel~6). In most of the theoretical studies ~0-~2.~4.17),the factorization of the two-time intensity correlation function has been considered to be a good approximation. It has been shown 12) that even when the pump field has a finite bandwidth, the two-time intensity correlation function for a two-level atom is factorizable into a product of the mean light intensities at the two times in question. On the other hand, in the presence of a chaotic pump field, the intensity correlation function cannot be factorized~S). Fluctuation effects on the intensity correlation function arising from the presence of many atoms ~9) as well as from the presence of the pump field have been considered in detail. We have recently considered 21) the excitation spectrum arising from photon density-density correlations in a system of two interacting identical atoms in the limit of high photon densities. In this model, the two-level atoms are pumped near resonance by a strong laser field while the atoms interact through their dipole-dipole interaction and radiate to each other as well. In such a system, photon density-density correlations arise from the interference spectrum due to the beating of the frequency of the pump field and the atomic transition frequency as well as from the spectrum induced by the dipole-dipole and radiative interactions between the atoms. The spectral function and the two-time photon density-density correlation function of the system have been calculated and discussed2~). We would like to point out that the spectral function of a system corresponding to the photon density-density correlations describes the low frequency spectrum of the system under investigation and it has peaks at the frequency modes, which are induced by the interactions or by the correlations occurring within the system. This implies that the photon density-density correlation spectrum describes the dynamics of the low frequency spectrum of the system, and it should be distinguished from the spectrum of the stationary solutions derived from the steady-state photon density fluctuations,
PHOTON DENSITY-DENSITY CORRELATION SPECTRUM
(coo-
205
(~a
w v
I
~
1
Fig. 1. Energy levels for an atom with two upper (excited) states and one lower (ground) state. Fig. 2. Energy levels for an atom with two lower (one excited and one ground) states and one upper (excited) state.
which occur at zero frequencies and indicate only static effects. Thus the study of the photon density--density correlation spectrum of a system is important since it gives an account of the low frequency spectrum of the system in question. Using the spectral function, one can also calculate the two-time photon density-density correlation function, which describes the behaviour of the system at two different times. The purpose of the present study is to investigate the excitation spectrum arising from photon density-density correlations at high photon densities of a single three-level atom, whose energy levels are depicted in figs. 1 and 2, respectively. In such a system, photon density-density correlations may be due to the interference spectrum arising from the beating between the two electronic transition frequencies of the atom. The model Hamiltonian is discussed in section 2 and it is used in section 3 to derive an expression for the photon density-density Green's function. In section 4, the Green's functions of the system have been calculated selfconsistently by making use of a decoupling approximation, which is applicable in the limit of high photon densities. The spectral function describing the excitation spectrum of the system has been calculated and discussed in section 5. Expressions for the photon density-density correlation function and for the mean square photon density distribution function have been derived in section 6, while the main results are summarized in section 7.
2. The Hamiltonian of the system
We consider a three-level atom, where the nondegenerate energy levels for the ground and two excited states are denoted by 1, 2 and 3 and the corresponding energies by to1, to2 and to3, respectively, to1 < to2 < to3 as shown in fig. 1. The atom is pumped by a strong electromagnetic field (pump field) whose energy mode toa is initially populated and it will be taken to be equal to oJ3~= tOa, where to3~= to3 - to~, to2~= to2 - to~ = toa- A and to32= A are the atomic transition frequencies of the levels in question. The electronic transitions 1~-~3 and 1.->2 are electric dipole allowed while the transition 2,~3 is
206
CONSTANTINE
MAVROYANNIS
forbidden. The energy levels 2 and 3 are supposed to be closely spaced so that 0)32= A is much less than 0)a. The transitions 1<-->3and 1~2, as shown in fig. 1 describe physical processes corresponding to one-photon resonance and near resonance, respectively. The atom is simultaneously coupled to the remaining modes of the electromagnetic field (signal field), those being initially unpopulated. The Hamiltonian for the atomic system shown in fig. 1 may be taken as ~a :
1" -$ t (0)a -- A ) n 2 + COati3 + 0)aria + 2 10)p~/f2(O~lO/2fla -- O~tOdl~a)
[
0)21
?
?
1/2
+ ~i0)p ~
? O? f,3(k, A ) 0)31] - ~ ] z(o~,a3~,,xa~a,/3~a),
(1)
where a~ and a~ are the Fermi creation and annihilation operators representing the electron states i = 1, 2, 3 and electron number operator ni = a~a~ while fla* and fla are the photon creation and annihilation operators of the pump field with frequency 0), and photon number operator na= fl*afl,. The coupling functions f2 and f3 ar e defined as f2=f12(0)21/0)a)=f12(1-A/0)a) and f3 = fl3(0)3J0)a) = ft3, where ft2 and f13 are the oscillator strengths for the allowed electronic transitions 1.-~2 and 1~3, respectively and top is the plasma frequency; the units with n = 1 are used throughout. Since the electronic transition 2.-~3 is optically forbidden, the corresponding oscillator strength f23 vanishes, f23 = 0. The photon creation and annihilation operators /3*kaand/3~x describe the frequency modes of the signal field with wavevector k, frequency ck and transverse polarization A( -- 1, 2). In writing eq. (1) we have taken into account the relation n~ + nE + n3 = 1. The first three terms in eq. (1) represent the free two atomic fields and the pump field respectively while the fourth and the fifth ones describe the coupling between them. The last three terms in eq. (1) describe the free signal field and its interaction with the atomic levels, respectively. The pump field with frequency 0)~ is initially populated and it will be considered as the strong field in comparison with the signal field which is taken as the weak perturbing field.
3. The photon density--density Green's function We introduce the Fourier transform of the photon density-density retarded double-time Green's function Dna(0))= ((na(t); n~(t'))); the properties of the Green's functions can be found elsewhere22"23). Using the Hamiltonian (1) we
PHOTON D E N S I T Y - D E N S I T Y CORRELATION SPECTRUM
207
derive the equation of motion for the Green's function D,a(tO) as Dna(tO) _-- itOp 2to (
na(t'))),
(2)
where the operator X123a(t) is defined as X123a(t) = "V/f2(O:~OL2~at + t~2%Otl/3a) + %v'/T3(Ot~O£3/3a~"d- d ~ l / 3 a ).
(3)
We next derive the equation of motion for the Green's function <> by operating 22'23) on its rhs with respect to the argument t' in the form (
1 ( X^ m ) - i__~//v 2~rto 2 \\z~'123a; X123a>>,
(4)
where X123 = V/F2(°fI°/2/3a ~"- Ot~O/I/3a)-{-%V/f3(O/IOf3/3a %- O/3%O~I/3a),
(5)
and the time arguments of the operators have been suppressed for convenience. Substituting eq. (4) into eq. (2), we have •
D.a(tO) = -
The expression contribution to term describes density-density
2
41~2(2123>+ 4to2\\ -~//Xl23~, X123a>>. "
(6)
(6) for D.a(a,) is exact; the first term represents the static D.~(to) and it is in general different from zero while the last dynamic effects arising from frequency dependent photon correlations• Neglecting the first term in eq. (6), we may write
... 2 O.I _ {,,,
h*:f~ 2
Ona(O))= ~[Gl2a'((o) -{-G2la((O)]-{-~[Gl3at((o) -k G31a(£0)],
(7)
where use has been made of the notation: Gl2a*(to) -*" X123a>>, -- ((Or' *10/2/3a,
G21a(tO) = (>,
(8a)
G l 3 a , ( ~ ) = <
G31a(tO) = <>.
(8b)
In deriving eq. (7) from eq. (6), we have neglected nondiagonal Green's functions of the form <>, , , as being much smaller than the diagonal ones defined by eqs. (8a) and (8b). The spectral function J~a(tO) is defined 23) as J~a(tO) = - 2 Im D,a(tO)n(to),
(9)
where n(to) = (e ~°' - 1)-' with/3 = 1/kaT, Kb is Boltzmann's constant, T is the absolute temperature and Im stands for the imaginary part of the quantity in question. The Green's functions Gna*(to), GI3a*(to) and G2la(tO), G3,Rto) describe physical processes where the atomic operators aIa2, a|a3 and a~a,,
208
CONSTANTINE MAVROYANNIS
ata~ emit and absorb one photon mode of the pump field, respectively. Thus, to obtain the expression for the spectral function J.a(to) of the system we have to calculate the expressions for the Green's functions GjEa,(to), Gl2a(to), Glaat(to) and Gala(to) by means of the Hamiltonian (1) and then substitute their imaginary parts into eq. (9).
4. Calculation of the Green's functions of the system
Using the Hamiltonian (1) we derive the equations of motion for the Green's functions G1Ea*(to), G21a(to), Glaa*(to) and G31a(to) as (d2 + A)G12a,(to)
(nl2T/n2)(12 -F ~a) "+ 21itop'~V/T3G32na(to) +
=
21itop~V/T2F21 ha(to), (10)
(d2 - -
A)G21a(to)
~ - ~2)(12+ tia) + 2i itopX/f3G23.a(to) + 21itopX/f2F21.a(to), (~,-
=
(ll) daGlaa*(to)
(~2~a)(~ + aa) + 21ito~X/~G2ana(to) + 21ito~X/~Fa,.a(to),
daG31a(to)
(12)
(al~#0(~ + aa) + ~ itooVf:G~:~a(to) + 12ito~X/gFa,.a(to),
(13)
d2aG2a~(to) = - iO2~v/12+ & G l a a ' ( t o ) - iOaX/2r+ fi~G21a(to),
(14)
da2Ga2na(to) = - iO2~v/12 + naG31a(to) - if~3~/21 + t~aGl2a*(to),
(15)
where the propagators d2 = d2(to), d3 = da(to), d2a = d2a(to) and da2 = da2(to) are defined as d2 = to - 2i3'2;
d3 = to -- 213'3,
d23 = to - A - 123'+,
(16)
3'+ = 3'2 + 3'3,
(17)
d32 = to + A - 123'+,
3"2 = 3"2(to) = t°2p ~ [,2(k, A )(to21/ck ) 2 k,~ (to - ck + toa)
(18)
and 3'3 = 3'3(to) can be obtained from eq. (18) by interchanging 2 and 3 (2~--~3). In eqs. (10)-(15), nl = ( O ~ l O t l ) , n 2 = (Ot~a2), ffa = (~ta~a) is average value of the photon density operator while 02 and Oa represent energy shifts defined as O 2 = t o 2~ 'i+fia); pJ12~,2
~2=to2~ ~i+tia), p./13~2
(19)
and use has been made of the definitions for the Green's functions F21na(tO) = ( ( ( n 2 - nl)( 1 + ha);
X123a)),
F31na(to) = ( ( ( n 3 - 711)(12+ ha),
X123a)),
G32na(to) =
((a~a2(12+ n~); X123~)) and G23,,(to)= ((ata3(12+ na); Xn3a)). In deriving eqs. (14)
PHOTON DENSITY-DENSITY CORRELATION SPECTRUM
209
and (15) we have made use of the following decoupling approximations ((O/~Ot3~tana; XI23a)) ~ 2/laGl3a*(tO),
(20a)
((O~2~O/lna~3a; XI23a)) ~ 2/laG21a(tO),
(20b)
((O/IO/2~atna; X123a)) ~ 2/laG12a*(t°),
(20c)
((c~otlna/3a; X123a))~ 2naG31a(tO),
(20d)
which are applicable in the limit of high photon density24-26), na>> 1. The Green's functions G23na(to) and G32na(to) describe the interaction of the atomic operators a tot3 and Ot]Ot2 with the photon density operator ha, while the Green's functions F2~aa(to) and Fal~a(to) designate the interaction between the difference of the electron number operators n 2 - n l and n 3 - n l with the photon density operator na of the photon field. In the same approximation, the equations of motion for the Green's functions F21na(to) and F3~n~(to) turn out to be (tO -- 3'2)FElna(tO) = - 2iO2X/21-T ffa[G12a)(tO) + G21a(tO)] -
iO3X/~ + ~a[G13a+(to)+ G31a(to)],
(21)
(tO -- 3'3)F31na(tO) = - 2 iO3V'21-~ ffa[Gl3a*(tO) + G31a(tO)]
- iOzX/x2+ tia[Gi2~+(tO)+ G21a(tO)].
(22)
Substituting eqs. (14), (15), (21) and (22) into eqs. (10)-(13), we obtain tO - 3'2 =
(tll--"2)(12+~a)+
2Ir
+ -0-~0-~(to 1 - -
3'2
0"]2 G2,a(tO) q-
tO - 3'2
~_2___~
2(
2)G13a(tO)
+ ~-32)G31a(to),
(23)
tO - 3'2 -(ri'~f12)-(21-ha)+
022 G,Ea+(tO)+2(~-%G31a(tO ) tO -- 3'2
0 a3{
(a3
1
+ ~_2~23)G~3a+(to),
1
)
to - 3'5 (n
(24)
(21+ n~) +
+-O-~(tO 1 - -
3'5
G3~a(tO) + tO - 3'5 2
+ ~23) G2,a(tO),
Gau+(tO) 3) (25)
210
CONSTANTINE MAVROYANNIS (d3
02
02
~d322) G3,a(co)
co - 3'3
(aj-2~'r-~3)~1 ~2 +
-
+_~_~(co
02
ha) + co - 3'3GI3a*(co) + 2
(ff_S2_O~
3)G21a(co)
1 + ~__32)Gi2a,(co)" -- 3'3
(26)
The solution of the set of the coupled equations (23)-(26) gives the required expressions for the Green's functions G~2a*(co), G21a(co), G13a,(co) and G31a(co), respectively. To avoid complicated expressions, we take the limit when 3'2 ~ 3'3 ~ 3' and /]2 ~ / ] 3 ~ / ] ; such an approximation is applicable since all the parameters in question are of the same order of magnitude. Then in this approximation, we solve eqs. (23)-(26) with the result GI2a*(co) + G21a(co) + G13a*(co) + G31a(co)
,~(~+tia)f - _ - 2-ff-~-(~--~[ 3 0 ~ ( n 2 - ~3) - 2 ( a ~ - h2)[(co - 3 ¢ 4 ) 2 - za2 - 02]
(o~-~~2j,
(27)
where the function D(co) is determined by D ( c o ) = [(co - 3'/2)(co - 3') - 3 / ] 2 - A 2][(co - 3')2 _ A 2 _ 2/]2] _ 2 / ] 4 _
3/]2A2
(28a) [(co - 3 ¢ 4 ) 2 - zl 2+][(co_ 3 ¢ 4 ) 2 _ A 2].
(28b)
The energy shifts A+ and zl_ are given by 5/]2
+ 3/]2/
4A 2
A 2 = ~ - - + ZI2_-~--~/1 + 3/]2.
(29)
Substituting eqs. (27) and (28b) into eq. (7), we get Dna(co) =
( A / ] 2/8 q'/'co2)
[(co - 3 ¢ 4 ) 2 - za~][(co - 3 ¢ 4 ) 2 - A 2_] f
X ~3/]2(h2 - ti3) -- 2(nl -- ti2)[(co -- 33'/4) 2 -- A2 - - / ] 2 ]
33'/4) 2
/]2j. (30)
The expression (30) for the Green's function D,a(co) will be used in the next section to discuss the excitation spectrum under investigation.
PHOTON DENSITY-DENSITY CORRELATION SPECTRUM
211
5. Excitation spectrum To determine the excitation spectrum, we calculate the imaginary part of the decay function 3'2(to) given by eq. (18) as
4[to21 ÷ 0.))2] I PIE 12= ~/to, ÷ ~toV Im 3'2(to) = ~[--~(toa ~---~'-~_ } V0,
(31)
where V0 = 3 4 ( t o 2 1 / c ) 3 I Pl212 with P12 being the transition dipole moment and 3'o is the spontaneous emission probability corgesponding to 1,-,2 transition. Similarly, t0 2^3'0, Im 73(to) -_4[to31tto ~ [ - - ~ a +to)2]le1312= (1__ ~aa)
(32)
where ~/o = ](toa/c) 3 [P~3 [2, toa = to3, and "~0 is the spontaneous emission probability corresponding to 1~3 transition. The ratio ImT2(to) = [Pi2 Im3'3(to) ~ 12(1 - Alto~)
(33)
implies that for Pt2 ~ Pl3 and A ~toa, it is sufficient to take Im3'2(to)~ Im3'3(to) -~ 3'0. The real parts of the functions 3'2(to) and 3'3(to) give very small energy shifts (Lamb shift) and will be discarded. Then eq. (30) may be rewritten as
1
-to
1 a A++4i3'o)
1
- U_(to- a_1+ 4ai3'o _ to + A_ +4~i3'o) 1 - *(to _ O1+ 4ai3'o- to + D + 4ai3'o)],
(34)
where the propability amplitudes U+, U- and 4) are determined by V'1//'rl± f-r u_+ = 21/±(1/+ - 1/_)j) n! - ~2)[2(1 + 1/- 1/_+)- 3] + (~., - ti3)[3 -
(~l - ~3)~ 3 $ = 2(1/+ - 1)(1/_ - 1)'
1/_+--
ill
(35) (36)
while the parameters 77, 1/+ and 1/_ are defined as 7) = A 2/f~ 2,
(37a)
1/± = A 2±//] 2 = ~2 + 7) -+ 2aX/ 1 + 41//3.
(37b)
Taking the imaginary part of eq. (34), we derive the spectral function Jn,(to)
212
CONSTANTINE MAVROYANNIS
defined by eq. (9), as
J.a(to)
=
U÷
U_
[
(to _
~0
a+)2 + (~/0)2- (to + a+)2 + dr0)2]
_
~/0
]
[(to - a_) 2 + (Iv0) ~ (to + A)~ + dv0)2J
- ,[(to
4a~/° dr0) 2 - (to + 0aY° ]~ - 0)2+ ) 2 + (~,0)JY
(38)
The spectral function J,a(to) given by eq. (38) describes the excitation spectrum of the photon density-density correlations of a three-level atom whose energy levels are depicted in fig. 1. The six terms in eq. (38) describe three pairs of Lorentzian lines peaked at the frequencies to = _+A+, to = _+A_ and to = -+ 0, respectively, and having radiative widths of the order of ]y0. Positive and negative terms in eq. (38) imply that the physical process of absorption and amplification may take place, respectively. The expressions (35) and (36) imply that the probability amplitudes U+, U_ and tb depend on the values of the parameters "0 and a~_+and since ~/+ and ~/_ can be expressed from eq. (37b) in terms of B, there is only the parameter 7/ which determines the amplitudes U_+and th. In table I, the computed values of U+, U_ and (b are listed for given values of B. The data in table I indicate that for values of a9 less than B <2.5, U+ is negative and vanishes for B > 2.5 while U_ is negative for B-<2, it vanishes for B ~2.25 and it becomes positive for values of a9-> 2.5. The sixth column in table I implies that the probability amplitude th is always positive and increases for increasing values of ~/, and it becomes equal to or larger than unity, $ -> 1, for values of B -> 2. The three pairs of bands described by eq. (38) are similar in form to the three pairs of sidebands arising from the interference effects (quantum beats) in the resonance fluorescence of a three-level atom27). Thus the spectral function (38) is analogous to that describing the interference spectrum for the same atom when the main peak at the frequency to = A is missing27). The difference is that the probability amplitudes for the interference spectra are porportional to (D/toa) 2, while the corresponding ones for the photon densitydensity correlations given by eq. (38) are proportional to (D/to) 2 at the frequencies to determined by to = A+, to = A- and to = D. Therefore, the interference spectrum, which describes nondiagonal effects, is of the second order in perturbation with a probability of occurrence 27) of the order of (O/toa)2, while the corresponding spectra arising from the photon densitydensity correlations may be of the first order since their probability amplitudes depend on the parameters such as (D/A+) 2, (O/A-) 2 and (D/D) 2= 1, which under favorable conditions may be of the order of unity. We have also calculated the excitation spectrum describing the photon
PHOTON DENSITY-DENSITY CORRELATION SPECTRUM
-~÷,~+ ÷ , ~ ÷ A
~ (",1
¢q p~
' ,...-
+
~
b
'
t - ~ - ~
L,
~
÷
g{
+
"~ o
7~
~ II
1 I ~I I ~I i
~I I I I I I
'
L
213
214
C O N S T A N T I N E MAVROYANNIS
density-density correlations of a three-level atom, where the electronic transitions take place from a common upper (excited) level and two different lower (one excited and one ground) levels as depicted in fig. 2. The electronic transitions l o 3 and 2 o 3 are electric dipole allowed while the l o 2 is forbidden. Using the appropriate Hamiltonian describing the electronic transition l o 3 and 2 ~ 3 as shown in fig. 2, which is analogous to the Hamiltonian (1), we have calculated the spectral function describing the photon densitydensity correlations. The procedure to calculate the spectral function is the same as that leading to eq. (38) and it will not be repeated. The spectral function, which describes the photon density-density correlations of an atom shown in fig. 2, is found to be determined by eq. (38) provided that 433,0is replaced everywhere by 2~T0. The photon density-density correlations spectrum is found to be analogous to the corresponding one arising from the interference effects in the resonance fluorescence of the three-level atom shown in fig. 2 when the main peak at ,0 -- A is missing, which in this case has a delta function distribution27). The discussion given before for eq. (38) is applicable to this case as well.
6. The density--density correlation function
Using the expression for the spectral function Jna(`0), we may derive 23) the photon density-density correlation function (na(t')na(t)) as +or
(na(t')na(t))
= f Jna(`0)e -i~ d`0,
(39)
-oo
where T = t - t' and for r = 0, the photon mean square density distribution function
(n2a(t)) = (n~)= f
Jna(to)d`0.
(4O)
In the limit when 3'o--"0, which is applicable whenever the laser field is very strong, then the spectral function, eq. (38), takes the form of delta-function distributions, i.e., J . a ( ` 0 ) = 4tn(`0){ U + [ 8 ( `0 - A+) - 8 ( , o + ~+)1 -
U_[8(`0 - A)
- 8 ( , 0 + ~ - ) 1 - 6 [ 8 ( , 0 - O ) - 8 ( , 0 + O)1},
(41)
PHOTON DENSITY-DENSITY CORRELATION SPECTRUM
215
for 3'0--' 0. Inserting eq. (38) into eq. (39) and after integrating over to, we have (n,(t')na(t)) = 41{U+[i sin(z/l+) + cos(~-A+)coth41/3A] - U_[i sin(~-A_) + cos(~-A_)coth~/3a ] - $ [i sin(~-O) + cos(TO)coth2t/30 ]}e-3Y°'d4.
(42)
In the limit when 7o ~ 0, the exponent ( - 3y0~-/4) in eq. (42) should be replaced by unity; the same expression can be obtained from eqs. (39) and (41) after the integration over to has ben performed. In the limit when ~- = t - t ' = 0, eq. (42) is reduced to the following expression describing the mean square photon density distribution function (na2) = ~,[U+ coth ½/3A+- U- coth ½/3A-- 4~ coth ½/30].
(43)
In eqs. (42) and (43), coth~/3A+, coth~/3A_ and coth2t/30 designate the thermal contributions arising from the energy modes a+, A and O, respectively. In the high and low frequency limits, we have (i) High frequency limit, when/3A_+ >> 1 a n d / 3 0 >> 1, coth~/3A± ~ 1 + 2 e -~a-~,
(44a)
coth12/3O ~ 1 + 2 e -~a.
(44b)
(ii) Low frequency limit, when /3A_+~ 1 a n d / 3 0 ,~ 1, coth~2/3A± ~ 1 + 2//3za± = 1 + 2kaT[A±,
(45a)
coth2t/30 ~ 1 + 2//30 = 1 + 2kBT[O,
(45b)
while at zero temperature, /3 ~ , reduced to 4(n 2) = U+ - U_ - ~,
cothX2/3A_+= coth21/30 = 1 and eq. (43) is
for/3 = oo.
(46)
Taking into account that the frequencies A, O, and consequently, A+ and A_ belong to the low range of the frequency spectrum, thermal effects may turn out to be important for the correlation as well as for the distribution functions given by eqs. (42) and (43), respectively. The first two terms in eqs. (42) and (43) may be positive or negative depending on the values of the parameter 71, while the last terms in both equations are always negative. The data in table I indicate that for values of the parameter ~->2, i.e., for A2->2O 2, the last term in eqs. (42) and (43) dominates and is always negative. Taking into consideration that h2 and fi3 are very small quantities indeed, i.e., h2 "~ 1 and h3 "~ 1, the last column in table I indicates that the expression for the mean square photon density distribution function (n~) determined by eq. (46) is positive for rt <0.5, it vanishes for values of ~q approximately between 0.7 > ~q > 0.5 and then it becomes negative
216
CONSTANTINE MAVROYANNIS
for ~1-0.75- Such a behaviour at zero temperature may indicate that for <0.5 the photons of the pump field are correlated, while for ~ > 0.7, the photons are not correlated and the photon anticorrelation or the photon antibunching effect takes place. Similar behaviour may be expected to occur for the photon density-density correlation function given by eq. (42) for T near zero, small 7o and /3 = oo. Numerical values at finite temperatures for different values of ~1 can be easily obtained from eq. (43), while for finite values 1" and 3'0, eq. (42), has to be taken into consideration. The expressions (42) and (43) can be easily evaluated for different values of the parameters ~!, V0, T and temperature T. The dependence of the expressions for the spectral function, the photon density-density correlation function and the mean square photon density distribution function on the parameter 7/ is an important property because it makes it possible to study experimentally the expressions in question by properly varying 7/ or to measure A by properly choosing the intensity of the laser field which defines 12.
7. Discussion
We have calculated the excitation spectrum arising from photon densitydensity correlations of a three-level atom whose active levels are depicted in figs. 1 and 2, respectively. Using a self-consistent approach, which is valid in the limit of high photon densities, the spectral function (38) has been derived for an atom whose energy level diagram is shown in fig. 1. The spectral function (38) describes three pairs of Lorentzian lines peaked at to = __A., to = _ A_ and to = - 12 and having spectral widths of the order of ~70- The probability amplitudes U+, U_ and $ for the occurrence of these three pairs of bands are given by eqs. (35) and (36), respectively, as a function of the parameter a~. Numerical values for the probability amplitudes for different values of ~ are listed in table I. The positive and negative terms in eq. (38) indicate that the physical processes of attenuation and amplification of the signal field may take place, respectively. The excitation spectrum described by the spectral function (38) is similar to that arising from the interference effects in the resonance fluorescence of a three-level atom when the central peak at to = A is missing27). They differ in that the photon density-density correlation spectrum is a first order effect while that of the interference one is of the second order in perturbation. The spectral function describing the photon density--density correlations of an atom shown in fig. 2 is also determined by eq. (38) if a4~/0 is replaced everywhere by ~70. The photon density-density correlation function at t # t' is determined by
PHOTON DENSITY-DENSITY CORRELATION SPECTRUM
217
eq. (42) as a function of ~- = t - t', ~/0 and temperature T. In the limit when ~/0-->0, then exp(-]~/0"r) in eq. (42) should be replaced by unity. At equal times when ~"= 0, eq. (42) is reduced to eq. (43), which describes the mean photon square density distribution function as function of temperature. It is shown that for values of the parameter ~->2, the expressions for the photon density-density correlation function for ~"~ 0 and the mean square photon density distribution function become negative indicating the phenomenon of the photon antibunching may be anticipated to occur. Since the frequency modes A+, A- and O are in the range of the low frequency spectrum, the temperature dependence of the density-density correlation function as well as the square of the photon density distribution function is expected to be substantial. Values for the probability amplitudes U+, U_, 4~ and for the square photon density distribution function (n 2) at zero temperature are listed in table I for different values of the parameter T/. The expression for the density-density correlation function given by eq. (42) can be easily computed for given values of the parameters 77, ~'0, ~" and temperature T. The density-density correlation function determined by eq. (42) may be measured experimentally by means of photoelectric-correlation experiments of the type carried out by Kimble et al) 5) and by Dagenais and MandeP6). Our treatment here is restricted to a single three-level atom while in the presence of a second similar atom, cooperative effects are anticipated to occur arising from the dipole-dipole and radiative interactions between the atoms; this will be the subject of a future publication. We hope that the present study will stimulate photoelectric-correlation experiments to investigate photon correlation effects in three-level atoms such as to verify and enlighten our theoretical predictions.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)
M.O. Scully and W.E. Lamb, Jr., Phys. Rev. 159 (1967) 208. L. Allen, C.Y. Huang and L. Mandel, Opt. Comm. 22 (1978) 251. C.Y. Huang and L. Mandel, Phys. Rev. AI8 (1978) 644. F. Casagrande and R. Cordoni, Phys. Rev. AI8 (1978) 1628. M.S. Zubairy, Phys. Rev. A20 (1979) 2464. R. Hanbury Brown and R.Q. Twiss, Nature (London) 177 (1956) 27; Proc. Roy. Soc. London A242 (1957) 300; A243 (1957) 291. R.J. Glauber, Phys. Rev. Lett. I0 (1963) 84; Phys. Rev. 130 (1963) 2529; 131 (1963) 2766. L. Mandel and E. Wolf, Rev. Mod. Phys. 37 (1965) 231. D. Stoler, Phys. Rev. Lett. 33 (1974) 1397. H.J. Carmichael and D.F. Walls, J. Phys. B9 (1976) 1199. H.J. Kimble and L. Mandel, Phys. Rev. AI3 (1976) 2123. H.J. Kimble and L. Mandel, Phys. Rev. AI5 (1977) 689.
218 13) 14) 15) 16) 17) 18) 19) 20)
CONSTANTINE MAVROYANNIS
M. Kozierowski and R. Tanfis, Opt. Comm. 21 (1977) 229. C. Cohen-Tannoudji and S. Reynaud, Phil. Trans. Roy. Soc. (London) A293 (1979) 223. H.J. Kimble, M. Dagenais and L. Mandel, Phys. Rev. Lett. 39 (1977) 691. M. Dagenais and L. Mandel, AI8 (1978) 2217. G.S. Agarnal, Phys. Rev. Ai5 (1977) 814. M. Schubert, K.E. Siisse and W. Vogel, Opt. Comm. 30 (1978) 275. H.J. Kimble, M. Dagenais and L. Mandel, Phys. Rev. AI8 (1978) 201. M. Schubert, K.E. Siisse, W. Vogel and D.G. Welsch, Optical and Quantum Electronics 12 (1980) 65. 21) C. Mavroyannis, Phys. Rev. A 22 (1980) 1129. 22) D.N. Zubarev, Usp. Fiz. Nauk 71 (1960) 71 (Sov. Phys. Usp. 3 (1960) 320). 23) C. Mavroyannis, in Physical Chemistry, Vol. XIA, ed. H. Eyring, W. Jost and D. Henderson (Academic Press, New York, 1975) p. 487. 24) C. Mavroyannis, Phys. Rev. AI8 (1978) 185. 25) C. Mavroyannis, Opt. Comm. 26 (1978) 453; 27 (1978) 371; 29 (1979) 80. 26) C. Mavroyannis, Physica A99 (1979) 435. 27) C. Mavroyannis, Can. J. Phys. 58 (1980) 957.
SPECTRUM OF A T H R E E - L E V E L ATOM AT HIGH PHOTON DENSITIES* Constantine MAVROYANNIS Division o[ Chemistry, National Research Council of Canada, Ottawa, Ontario, Canada K 1A OR6 Received 2 July 1980
The excitation spectrum arising from photon density--density correlations has been calculated for two types of a single three-level atom for which electronic transitions occur: from two different upper levels to a common lower one and from a common upper level to two different lower levels, respectively. Photon density-density correlations in such a system are described by the interference spectrum, which results from the beating between the two electronic transition frequencies. The spectral function has been calculated in the limit of high photon densities and consists of three pairs of bands, which are similar to those of the interference spectrum when the main peak at the frequency to = A is missing, where A is equal to the frequency splitting between the excited states of the atom. The probability amplitude for the occurrence of each pair of bands depends on the parameter 7/= A2/02, where/] is the Rabi frequency. Numerical values indicate that the probability amplitudes become negative, zero, or positive for different values of the parameter 7. Using the spectral function, an expression has has been derived for the photon density-density correlation function, which describes photon density-density correlations at two different times and at finite temperatures. In the limit of equal times, the mean square photon density distribution function has been calculated and discussed. At zero temperatures, numerical values for the mean square photon density distribution function indicate that it is positive for "11<0.5, it vanishes for values of rt approximately 0.7> "0 >0.5, and it becomes negative for "0 - 0.75. The possibility of observing photon density-density correlations is discussed.
1. Introduction T h e r e has b e e n c o n s i d e r a b l e i n t e r e s t in the f l u c t u a t i o n s p e c t r a of laser beams~-5). M o s t of the i n v e s t i g a t i o n s h a v e b e e n d e v o t e d to s t u d y i n g the effects of c o o p e r a t i v e a n d a t o m i c i n t e r a c t i o n s o n the p t h o t o n statistics of a single m o d e laser. E x p r e s s i o n s for the p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n of the p h o t o n o c c u p a t i o n n u m b e r as well as e x p e c t a t i o n v a l u e s of the light i n t e n s i t y a n d i n t e n s i t y c o r r e l a t i o n s h a v e b e e n d e r i v e d b y m a k i n g use of different m a t h e m a t i c a l t r e a t m e n t s ; we r e f e r to a r e c e n t p a p e r b y Zubairy5), w h e r e details a n d r e f e r e n c e s c a n b e f o u n d . T h e p h o t o n c o r r e l a t i o n or p h o t o n b u n c h i n g p h e n o m e n o n has b e e n first d e m o n s t r a t e d b y H a n b u r y B r o w n a n d T w i s s 6) a n d d e s c r i b e s the d i s t i n c t t e n d e n c y o f p h o t o n s in a light b e a m e m i t t e d b y a t h e r m a l e q u i l i b r i u m s o u r c e * Issued as NRCC No. 18708. 203
204
CONSTANTINE MAVROYANNIS
to arrive at a detector in bunches or correlated pairs. The bunching effect has been studied implicitly in time-resolved correlation experiments, where photon field correlations have been measured with the help of photoelectric detectors. Glauber 7) has shown that the joint probability density of photodetection is proportional to the normally ordered correlation function, which is of the fourth order in the field operators, or of the second order in the light intensity; we refer to the review paper by Mandel and Wolf a) where a detailed discussion and literature are given. The photon antibunching effect or the anticorrelation effect in the resonance fluorescence from a single two-level atom has been discussed by Stolerg), Carmichaei and Wallsl°), Kemble and Mandelll'12), Kozierowski and Tanfisl3), Cohen-Tannoudji and Reynaud ~4) and was observed first by Kimble et al. ~5) and later by Degenais and Mandel~6). In most of the theoretical studies ~0-~2.~4.17),the factorization of the two-time intensity correlation function has been considered to be a good approximation. It has been shown 12) that even when the pump field has a finite bandwidth, the two-time intensity correlation function for a two-level atom is factorizable into a product of the mean light intensities at the two times in question. On the other hand, in the presence of a chaotic pump field, the intensity correlation function cannot be factorized~S). Fluctuation effects on the intensity correlation function arising from the presence of many atoms ~9) as well as from the presence of the pump field have been considered in detail. We have recently considered 21) the excitation spectrum arising from photon density-density correlations in a system of two interacting identical atoms in the limit of high photon densities. In this model, the two-level atoms are pumped near resonance by a strong laser field while the atoms interact through their dipole-dipole interaction and radiate to each other as well. In such a system, photon density-density correlations arise from the interference spectrum due to the beating of the frequency of the pump field and the atomic transition frequency as well as from the spectrum induced by the dipole-dipole and radiative interactions between the atoms. The spectral function and the two-time photon density-density correlation function of the system have been calculated and discussed2~). We would like to point out that the spectral function of a system corresponding to the photon density-density correlations describes the low frequency spectrum of the system under investigation and it has peaks at the frequency modes, which are induced by the interactions or by the correlations occurring within the system. This implies that the photon density-density correlation spectrum describes the dynamics of the low frequency spectrum of the system, and it should be distinguished from the spectrum of the stationary solutions derived from the steady-state photon density fluctuations,
PHOTON DENSITY-DENSITY CORRELATION SPECTRUM
(coo-
205
(~a
w v
I
~
1
Fig. 1. Energy levels for an atom with two upper (excited) states and one lower (ground) state. Fig. 2. Energy levels for an atom with two lower (one excited and one ground) states and one upper (excited) state.
which occur at zero frequencies and indicate only static effects. Thus the study of the photon density--density correlation spectrum of a system is important since it gives an account of the low frequency spectrum of the system in question. Using the spectral function, one can also calculate the two-time photon density-density correlation function, which describes the behaviour of the system at two different times. The purpose of the present study is to investigate the excitation spectrum arising from photon density-density correlations at high photon densities of a single three-level atom, whose energy levels are depicted in figs. 1 and 2, respectively. In such a system, photon density-density correlations may be due to the interference spectrum arising from the beating between the two electronic transition frequencies of the atom. The model Hamiltonian is discussed in section 2 and it is used in section 3 to derive an expression for the photon density-density Green's function. In section 4, the Green's functions of the system have been calculated selfconsistently by making use of a decoupling approximation, which is applicable in the limit of high photon densities. The spectral function describing the excitation spectrum of the system has been calculated and discussed in section 5. Expressions for the photon density-density correlation function and for the mean square photon density distribution function have been derived in section 6, while the main results are summarized in section 7.
2. The Hamiltonian of the system
We consider a three-level atom, where the nondegenerate energy levels for the ground and two excited states are denoted by 1, 2 and 3 and the corresponding energies by to1, to2 and to3, respectively, to1 < to2 < to3 as shown in fig. 1. The atom is pumped by a strong electromagnetic field (pump field) whose energy mode toa is initially populated and it will be taken to be equal to oJ3~= tOa, where to3~= to3 - to~, to2~= to2 - to~ = toa- A and to32= A are the atomic transition frequencies of the levels in question. The electronic transitions 1~-~3 and 1.->2 are electric dipole allowed while the transition 2,~3 is
206
CONSTANTINE
MAVROYANNIS
forbidden. The energy levels 2 and 3 are supposed to be closely spaced so that 0)32= A is much less than 0)a. The transitions 1<-->3and 1~2, as shown in fig. 1 describe physical processes corresponding to one-photon resonance and near resonance, respectively. The atom is simultaneously coupled to the remaining modes of the electromagnetic field (signal field), those being initially unpopulated. The Hamiltonian for the atomic system shown in fig. 1 may be taken as ~a :
1" -$ t (0)a -- A ) n 2 + COati3 + 0)aria + 2 10)p~/f2(O~lO/2fla -- O~tOdl~a)
[
0)21
?
?
1/2
+ ~i0)p ~
? O? f,3(k, A ) 0)31] - ~ ] z(o~,a3~,,xa~a,/3~a),
(1)
where a~ and a~ are the Fermi creation and annihilation operators representing the electron states i = 1, 2, 3 and electron number operator ni = a~a~ while fla* and fla are the photon creation and annihilation operators of the pump field with frequency 0), and photon number operator na= fl*afl,. The coupling functions f2 and f3 ar e defined as f2=f12(0)21/0)a)=f12(1-A/0)a) and f3 = fl3(0)3J0)a) = ft3, where ft2 and f13 are the oscillator strengths for the allowed electronic transitions 1.-~2 and 1~3, respectively and top is the plasma frequency; the units with n = 1 are used throughout. Since the electronic transition 2.-~3 is optically forbidden, the corresponding oscillator strength f23 vanishes, f23 = 0. The photon creation and annihilation operators /3*kaand/3~x describe the frequency modes of the signal field with wavevector k, frequency ck and transverse polarization A( -- 1, 2). In writing eq. (1) we have taken into account the relation n~ + nE + n3 = 1. The first three terms in eq. (1) represent the free two atomic fields and the pump field respectively while the fourth and the fifth ones describe the coupling between them. The last three terms in eq. (1) describe the free signal field and its interaction with the atomic levels, respectively. The pump field with frequency 0)~ is initially populated and it will be considered as the strong field in comparison with the signal field which is taken as the weak perturbing field.
3. The photon density--density Green's function We introduce the Fourier transform of the photon density-density retarded double-time Green's function Dna(0))= ((na(t); n~(t'))); the properties of the Green's functions can be found elsewhere22"23). Using the Hamiltonian (1) we
PHOTON D E N S I T Y - D E N S I T Y CORRELATION SPECTRUM
207
derive the equation of motion for the Green's function D,a(tO) as Dna(tO) _-- itOp 2to (
na(t'))),
(2)
where the operator X123a(t) is defined as X123a(t) = "V/f2(O:~OL2~at + t~2%Otl/3a) + %v'/T3(Ot~O£3/3a~"d- d ~ l / 3 a ).
(3)
We next derive the equation of motion for the Green's function <
1 ( X^ m ) - i__~//v 2~rto 2 \\z~'123a; X123a>>,
(4)
where X123 = V/F2(°fI°/2/3a ~"- Ot~O/I/3a)-{-%V/f3(O/IOf3/3a %- O/3%O~I/3a),
(5)
and the time arguments of the operators have been suppressed for convenience. Substituting eq. (4) into eq. (2), we have •
D.a(tO) = -
The expression contribution to term describes density-density
2
41~2(2123>+ 4to2\\ -~//Xl23~, X123a>>. "
(6)
(6) for D.a(a,) is exact; the first term represents the static D.~(to) and it is in general different from zero while the last dynamic effects arising from frequency dependent photon correlations• Neglecting the first term in eq. (6), we may write
... 2 O.I _ {,,,
h*:f~ 2
Ona(O))= ~[Gl2a'((o) -{-G2la((O)]-{-~[Gl3at((o) -k G31a(£0)],
(7)
where use has been made of the notation: Gl2a*(to) -*" X123a>>, -- ((Or' *10/2/3a,
G21a(tO) = (
(8a)
G l 3 a , ( ~ ) = <
G31a(tO) = <
(8b)
In deriving eq. (7) from eq. (6), we have neglected nondiagonal Green's functions of the form <
(9)
where n(to) = (e ~°' - 1)-' with/3 = 1/kaT, Kb is Boltzmann's constant, T is the absolute temperature and Im stands for the imaginary part of the quantity in question. The Green's functions Gna*(to), GI3a*(to) and G2la(tO), G3,Rto) describe physical processes where the atomic operators aIa2, a|a3 and a~a,,
208
CONSTANTINE MAVROYANNIS
ata~ emit and absorb one photon mode of the pump field, respectively. Thus, to obtain the expression for the spectral function J.a(to) of the system we have to calculate the expressions for the Green's functions GjEa,(to), Gl2a(to), Glaat(to) and Gala(to) by means of the Hamiltonian (1) and then substitute their imaginary parts into eq. (9).
4. Calculation of the Green's functions of the system
Using the Hamiltonian (1) we derive the equations of motion for the Green's functions G1Ea*(to), G21a(to), Glaa*(to) and G31a(to) as (d2 + A)G12a,(to)
(nl2T/n2)(12 -F ~a) "+ 21itop'~V/T3G32na(to) +
=
21itop~V/T2F21 ha(to), (10)
(d2 - -
A)G21a(to)
~ - ~2)(12+ tia) + 2i itopX/f3G23.a(to) + 21itopX/f2F21.a(to), (~,-
=
(ll) daGlaa*(to)
(~2~a)(~ + aa) + 21ito~X/~G2ana(to) + 21ito~X/~Fa,.a(to),
daG31a(to)
(12)
(al~#0(~ + aa) + ~ itooVf:G~:~a(to) + 12ito~X/gFa,.a(to),
(13)
d2aG2a~(to) = - iO2~v/12+ & G l a a ' ( t o ) - iOaX/2r+ fi~G21a(to),
(14)
da2Ga2na(to) = - iO2~v/12 + naG31a(to) - if~3~/21 + t~aGl2a*(to),
(15)
where the propagators d2 = d2(to), d3 = da(to), d2a = d2a(to) and da2 = da2(to) are defined as d2 = to - 2i3'2;
d3 = to -- 213'3,
d23 = to - A - 123'+,
(16)
3'+ = 3'2 + 3'3,
(17)
d32 = to + A - 123'+,
3"2 = 3"2(to) = t°2p ~ [,2(k, A )(to21/ck ) 2 k,~ (to - ck + toa)
(18)
and 3'3 = 3'3(to) can be obtained from eq. (18) by interchanging 2 and 3 (2~--~3). In eqs. (10)-(15), nl = ( O ~ l O t l ) , n 2 = (Ot~a2), ffa = (~ta~a) is average value of the photon density operator while 02 and Oa represent energy shifts defined as O 2 = t o 2~ 'i+fia); pJ12~,2
~2=to2~ ~i+tia), p./13~2
(19)
and use has been made of the definitions for the Green's functions F21na(tO) = ( ( ( n 2 - nl)( 1 + ha);
X123a)),
F31na(to) = ( ( ( n 3 - 711)(12+ ha),
X123a)),
G32na(to) =
((a~a2(12+ n~); X123~)) and G23,,(to)= ((ata3(12+ na); Xn3a)). In deriving eqs. (14)
PHOTON DENSITY-DENSITY CORRELATION SPECTRUM
209
and (15) we have made use of the following decoupling approximations ((O/~Ot3~tana; XI23a)) ~ 2/laGl3a*(tO),
(20a)
((O~2~O/lna~3a; XI23a)) ~ 2/laG21a(tO),
(20b)
((O/IO/2~atna; X123a)) ~ 2/laG12a*(t°),
(20c)
((c~otlna/3a; X123a))~ 2naG31a(tO),
(20d)
which are applicable in the limit of high photon density24-26), na>> 1. The Green's functions G23na(to) and G32na(to) describe the interaction of the atomic operators a tot3 and Ot]Ot2 with the photon density operator ha, while the Green's functions F2~aa(to) and Fal~a(to) designate the interaction between the difference of the electron number operators n 2 - n l and n 3 - n l with the photon density operator na of the photon field. In the same approximation, the equations of motion for the Green's functions F21na(to) and F3~n~(to) turn out to be (tO -- 3'2)FElna(tO) = - 2iO2X/21-T ffa[G12a)(tO) + G21a(tO)] -
iO3X/~ + ~a[G13a+(to)+ G31a(to)],
(21)
(tO -- 3'3)F31na(tO) = - 2 iO3V'21-~ ffa[Gl3a*(tO) + G31a(tO)]
- iOzX/x2+ tia[Gi2~+(tO)+ G21a(tO)].
(22)
Substituting eqs. (14), (15), (21) and (22) into eqs. (10)-(13), we obtain tO - 3'2 =
(tll--"2)(12+~a)+
2Ir
+ -0-~0-~(to 1 - -
3'2
0"]2 G2,a(tO) q-
tO - 3'2
~_2___~
2(
2)G13a(tO)
+ ~-32)G31a(to),
(23)
tO - 3'2 -(ri'~f12)-(21-ha)+
022 G,Ea+(tO)+2(~-%G31a(tO ) tO -- 3'2
0 a3{
(a3
1
+ ~_2~23)G~3a+(to),
1
)
to - 3'5 (n
(24)
(21+ n~) +
+-O-~(tO 1 - -
3'5
G3~a(tO) + tO - 3'5 2
+ ~23) G2,a(tO),
Gau+(tO) 3) (25)
210
CONSTANTINE MAVROYANNIS (d3
02
02
~d322) G3,a(co)
co - 3'3
(aj-2~'r-~3)~1 ~2 +
-
+_~_~(co
02
ha) + co - 3'3GI3a*(co) + 2
(ff_S2_O~
3)G21a(co)
1 + ~__32)Gi2a,(co)" -- 3'3
(26)
The solution of the set of the coupled equations (23)-(26) gives the required expressions for the Green's functions G~2a*(co), G21a(co), G13a,(co) and G31a(co), respectively. To avoid complicated expressions, we take the limit when 3'2 ~ 3'3 ~ 3' and /]2 ~ / ] 3 ~ / ] ; such an approximation is applicable since all the parameters in question are of the same order of magnitude. Then in this approximation, we solve eqs. (23)-(26) with the result GI2a*(co) + G21a(co) + G13a*(co) + G31a(co)
,~(~+tia)f - _ - 2-ff-~-(~--~[ 3 0 ~ ( n 2 - ~3) - 2 ( a ~ - h2)[(co - 3 ¢ 4 ) 2 - za2 - 02]
(o~-~~2j,
(27)
where the function D(co) is determined by D ( c o ) = [(co - 3'/2)(co - 3') - 3 / ] 2 - A 2][(co - 3')2 _ A 2 _ 2/]2] _ 2 / ] 4 _
3/]2A2
(28a) [(co - 3 ¢ 4 ) 2 - zl 2+][(co_ 3 ¢ 4 ) 2 _ A 2].
(28b)
The energy shifts A+ and zl_ are given by 5/]2
+ 3/]2/
4A 2
A 2 = ~ - - + ZI2_-~--~/1 + 3/]2.
(29)
Substituting eqs. (27) and (28b) into eq. (7), we get Dna(co) =
( A / ] 2/8 q'/'co2)
[(co - 3 ¢ 4 ) 2 - za~][(co - 3 ¢ 4 ) 2 - A 2_] f
X ~3/]2(h2 - ti3) -- 2(nl -- ti2)[(co -- 33'/4) 2 -- A2 - - / ] 2 ]
33'/4) 2
/]2j. (30)
The expression (30) for the Green's function D,a(co) will be used in the next section to discuss the excitation spectrum under investigation.
PHOTON DENSITY-DENSITY CORRELATION SPECTRUM
211
5. Excitation spectrum To determine the excitation spectrum, we calculate the imaginary part of the decay function 3'2(to) given by eq. (18) as
4[to21 ÷ 0.))2] I PIE 12= ~/to, ÷ ~toV Im 3'2(to) = ~[--~(toa ~---~'-~_ } V0,
(31)
where V0 = 3 4 ( t o 2 1 / c ) 3 I Pl212 with P12 being the transition dipole moment and 3'o is the spontaneous emission probability corgesponding to 1,-,2 transition. Similarly, t0 2^3'0, Im 73(to) -_4[to31tto ~ [ - - ~ a +to)2]le1312= (1__ ~aa)
(32)
where ~/o = ](toa/c) 3 [P~3 [2, toa = to3, and "~0 is the spontaneous emission probability corresponding to 1~3 transition. The ratio ImT2(to) = [Pi2 Im3'3(to) ~ 12(1 - Alto~)
(33)
implies that for Pt2 ~ Pl3 and A ~toa, it is sufficient to take Im3'2(to)~ Im3'3(to) -~ 3'0. The real parts of the functions 3'2(to) and 3'3(to) give very small energy shifts (Lamb shift) and will be discarded. Then eq. (30) may be rewritten as
1
-to
1 a A++4i3'o)
1
- U_(to- a_1+ 4ai3'o _ to + A_ +4~i3'o) 1 - *(to _ O1+ 4ai3'o- to + D + 4ai3'o)],
(34)
where the propability amplitudes U+, U- and 4) are determined by V'1//'rl± f-r u_+ = 21/±(1/+ - 1/_)j) n! - ~2)[2(1 + 1/- 1/_+)- 3] + (~., - ti3)[3 -
(~l - ~3)~ 3 $ = 2(1/+ - 1)(1/_ - 1)'
1/_+--
ill
(35) (36)
while the parameters 77, 1/+ and 1/_ are defined as 7) = A 2/f~ 2,
(37a)
1/± = A 2±//] 2 = ~2 + 7) -+ 2aX/ 1 + 41//3.
(37b)
Taking the imaginary part of eq. (34), we derive the spectral function Jn,(to)
212
CONSTANTINE MAVROYANNIS
defined by eq. (9), as
J.a(to)
=
U÷
U_
[
(to _
~0
a+)2 + (~/0)2- (to + a+)2 + dr0)2]
_
~/0
]
[(to - a_) 2 + (Iv0) ~ (to + A)~ + dv0)2J
- ,[(to
4a~/° dr0) 2 - (to + 0aY° ]~ - 0)2+ ) 2 + (~,0)JY
(38)
The spectral function J,a(to) given by eq. (38) describes the excitation spectrum of the photon density-density correlations of a three-level atom whose energy levels are depicted in fig. 1. The six terms in eq. (38) describe three pairs of Lorentzian lines peaked at the frequencies to = _+A+, to = _+A_ and to = -+ 0, respectively, and having radiative widths of the order of ]y0. Positive and negative terms in eq. (38) imply that the physical process of absorption and amplification may take place, respectively. The expressions (35) and (36) imply that the probability amplitudes U+, U_ and tb depend on the values of the parameters "0 and a~_+and since ~/+ and ~/_ can be expressed from eq. (37b) in terms of B, there is only the parameter 7/ which determines the amplitudes U_+and th. In table I, the computed values of U+, U_ and (b are listed for given values of B. The data in table I indicate that for values of a9 less than B <2.5, U+ is negative and vanishes for B > 2.5 while U_ is negative for B-<2, it vanishes for B ~2.25 and it becomes positive for values of a9-> 2.5. The sixth column in table I implies that the probability amplitude th is always positive and increases for increasing values of ~/, and it becomes equal to or larger than unity, $ -> 1, for values of B -> 2. The three pairs of bands described by eq. (38) are similar in form to the three pairs of sidebands arising from the interference effects (quantum beats) in the resonance fluorescence of a three-level atom27). Thus the spectral function (38) is analogous to that describing the interference spectrum for the same atom when the main peak at the frequency to = A is missing27). The difference is that the probability amplitudes for the interference spectra are porportional to (D/toa) 2, while the corresponding ones for the photon densitydensity correlations given by eq. (38) are proportional to (D/to) 2 at the frequencies to determined by to = A+, to = A- and to = D. Therefore, the interference spectrum, which describes nondiagonal effects, is of the second order in perturbation with a probability of occurrence 27) of the order of (O/toa)2, while the corresponding spectra arising from the photon densitydensity correlations may be of the first order since their probability amplitudes depend on the parameters such as (D/A+) 2, (O/A-) 2 and (D/D) 2= 1, which under favorable conditions may be of the order of unity. We have also calculated the excitation spectrum describing the photon
PHOTON DENSITY-DENSITY CORRELATION SPECTRUM
-~÷,~+ ÷ , ~ ÷ A
~ (",1
¢q p~
' ,...-
+
~
b
'
t - ~ - ~
L,
~
÷
g{
+
"~ o
7~
~ II
1 I ~I I ~I i
~I I I I I I
'
L
213
214
C O N S T A N T I N E MAVROYANNIS
density-density correlations of a three-level atom, where the electronic transitions take place from a common upper (excited) level and two different lower (one excited and one ground) levels as depicted in fig. 2. The electronic transitions l o 3 and 2 o 3 are electric dipole allowed while the l o 2 is forbidden. Using the appropriate Hamiltonian describing the electronic transition l o 3 and 2 ~ 3 as shown in fig. 2, which is analogous to the Hamiltonian (1), we have calculated the spectral function describing the photon densitydensity correlations. The procedure to calculate the spectral function is the same as that leading to eq. (38) and it will not be repeated. The spectral function, which describes the photon density-density correlations of an atom shown in fig. 2, is found to be determined by eq. (38) provided that 433,0is replaced everywhere by 2~T0. The photon density-density correlations spectrum is found to be analogous to the corresponding one arising from the interference effects in the resonance fluorescence of the three-level atom shown in fig. 2 when the main peak at ,0 -- A is missing, which in this case has a delta function distribution27). The discussion given before for eq. (38) is applicable to this case as well.
6. The density--density correlation function
Using the expression for the spectral function Jna(`0), we may derive 23) the photon density-density correlation function (na(t')na(t)) as +or
(na(t')na(t))
= f Jna(`0)e -i~ d`0,
(39)
-oo
where T = t - t' and for r = 0, the photon mean square density distribution function
(n2a(t)) = (n~)= f
Jna(to)d`0.
(4O)
In the limit when 3'o--"0, which is applicable whenever the laser field is very strong, then the spectral function, eq. (38), takes the form of delta-function distributions, i.e., J . a ( ` 0 ) = 4tn(`0){ U + [ 8 ( `0 - A+) - 8 ( , o + ~+)1 -
U_[8(`0 - A)
- 8 ( , 0 + ~ - ) 1 - 6 [ 8 ( , 0 - O ) - 8 ( , 0 + O)1},
(41)
PHOTON DENSITY-DENSITY CORRELATION SPECTRUM
215
for 3'0--' 0. Inserting eq. (38) into eq. (39) and after integrating over to, we have (n,(t')na(t)) = 41{U+[i sin(z/l+) + cos(~-A+)coth41/3A] - U_[i sin(~-A_) + cos(~-A_)coth~/3a ] - $ [i sin(~-O) + cos(TO)coth2t/30 ]}e-3Y°'d4.
(42)
In the limit when 7o ~ 0, the exponent ( - 3y0~-/4) in eq. (42) should be replaced by unity; the same expression can be obtained from eqs. (39) and (41) after the integration over to has ben performed. In the limit when ~- = t - t ' = 0, eq. (42) is reduced to the following expression describing the mean square photon density distribution function (na2) = ~,[U+ coth ½/3A+- U- coth ½/3A-- 4~ coth ½/30].
(43)
In eqs. (42) and (43), coth~/3A+, coth~/3A_ and coth2t/30 designate the thermal contributions arising from the energy modes a+, A and O, respectively. In the high and low frequency limits, we have (i) High frequency limit, when/3A_+ >> 1 a n d / 3 0 >> 1, coth~/3A± ~ 1 + 2 e -~a-~,
(44a)
coth12/3O ~ 1 + 2 e -~a.
(44b)
(ii) Low frequency limit, when /3A_+~ 1 a n d / 3 0 ,~ 1, coth~2/3A± ~ 1 + 2//3za± = 1 + 2kaT[A±,
(45a)
coth2t/30 ~ 1 + 2//30 = 1 + 2kBT[O,
(45b)
while at zero temperature, /3 ~ , reduced to 4(n 2) = U+ - U_ - ~,
cothX2/3A_+= coth21/30 = 1 and eq. (43) is
for/3 = oo.
(46)
Taking into account that the frequencies A, O, and consequently, A+ and A_ belong to the low range of the frequency spectrum, thermal effects may turn out to be important for the correlation as well as for the distribution functions given by eqs. (42) and (43), respectively. The first two terms in eqs. (42) and (43) may be positive or negative depending on the values of the parameter 71, while the last terms in both equations are always negative. The data in table I indicate that for values of the parameter ~->2, i.e., for A2->2O 2, the last term in eqs. (42) and (43) dominates and is always negative. Taking into consideration that h2 and fi3 are very small quantities indeed, i.e., h2 "~ 1 and h3 "~ 1, the last column in table I indicates that the expression for the mean square photon density distribution function (n~) determined by eq. (46) is positive for rt <0.5, it vanishes for values of ~q approximately between 0.7 > ~q > 0.5 and then it becomes negative
216
CONSTANTINE MAVROYANNIS
for ~1-0.75- Such a behaviour at zero temperature may indicate that for <0.5 the photons of the pump field are correlated, while for ~ > 0.7, the photons are not correlated and the photon anticorrelation or the photon antibunching effect takes place. Similar behaviour may be expected to occur for the photon density-density correlation function given by eq. (42) for T near zero, small 7o and /3 = oo. Numerical values at finite temperatures for different values of ~1 can be easily obtained from eq. (43), while for finite values 1" and 3'0, eq. (42), has to be taken into consideration. The expressions (42) and (43) can be easily evaluated for different values of the parameters ~!, V0, T and temperature T. The dependence of the expressions for the spectral function, the photon density-density correlation function and the mean square photon density distribution function on the parameter 7/ is an important property because it makes it possible to study experimentally the expressions in question by properly varying 7/ or to measure A by properly choosing the intensity of the laser field which defines 12.
7. Discussion
We have calculated the excitation spectrum arising from photon densitydensity correlations of a three-level atom whose active levels are depicted in figs. 1 and 2, respectively. Using a self-consistent approach, which is valid in the limit of high photon densities, the spectral function (38) has been derived for an atom whose energy level diagram is shown in fig. 1. The spectral function (38) describes three pairs of Lorentzian lines peaked at to = __A., to = _ A_ and to = - 12 and having spectral widths of the order of ~70- The probability amplitudes U+, U_ and $ for the occurrence of these three pairs of bands are given by eqs. (35) and (36), respectively, as a function of the parameter a~. Numerical values for the probability amplitudes for different values of ~ are listed in table I. The positive and negative terms in eq. (38) indicate that the physical processes of attenuation and amplification of the signal field may take place, respectively. The excitation spectrum described by the spectral function (38) is similar to that arising from the interference effects in the resonance fluorescence of a three-level atom when the central peak at to = A is missing27). They differ in that the photon density-density correlation spectrum is a first order effect while that of the interference one is of the second order in perturbation. The spectral function describing the photon density--density correlations of an atom shown in fig. 2 is also determined by eq. (38) if a4~/0 is replaced everywhere by ~70. The photon density-density correlation function at t # t' is determined by
PHOTON DENSITY-DENSITY CORRELATION SPECTRUM
217
eq. (42) as a function of ~- = t - t', ~/0 and temperature T. In the limit when ~/0-->0, then exp(-]~/0"r) in eq. (42) should be replaced by unity. At equal times when ~"= 0, eq. (42) is reduced to eq. (43), which describes the mean photon square density distribution function as function of temperature. It is shown that for values of the parameter ~->2, the expressions for the photon density-density correlation function for ~"~ 0 and the mean square photon density distribution function become negative indicating the phenomenon of the photon antibunching may be anticipated to occur. Since the frequency modes A+, A- and O are in the range of the low frequency spectrum, the temperature dependence of the density-density correlation function as well as the square of the photon density distribution function is expected to be substantial. Values for the probability amplitudes U+, U_, 4~ and for the square photon density distribution function (n 2) at zero temperature are listed in table I for different values of the parameter T/. The expression for the density-density correlation function given by eq. (42) can be easily computed for given values of the parameters 77, ~'0, ~" and temperature T. The density-density correlation function determined by eq. (42) may be measured experimentally by means of photoelectric-correlation experiments of the type carried out by Kimble et al) 5) and by Dagenais and MandeP6). Our treatment here is restricted to a single three-level atom while in the presence of a second similar atom, cooperative effects are anticipated to occur arising from the dipole-dipole and radiative interactions between the atoms; this will be the subject of a future publication. We hope that the present study will stimulate photoelectric-correlation experiments to investigate photon correlation effects in three-level atoms such as to verify and enlighten our theoretical predictions.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)
M.O. Scully and W.E. Lamb, Jr., Phys. Rev. 159 (1967) 208. L. Allen, C.Y. Huang and L. Mandel, Opt. Comm. 22 (1978) 251. C.Y. Huang and L. Mandel, Phys. Rev. AI8 (1978) 644. F. Casagrande and R. Cordoni, Phys. Rev. AI8 (1978) 1628. M.S. Zubairy, Phys. Rev. A20 (1979) 2464. R. Hanbury Brown and R.Q. Twiss, Nature (London) 177 (1956) 27; Proc. Roy. Soc. London A242 (1957) 300; A243 (1957) 291. R.J. Glauber, Phys. Rev. Lett. I0 (1963) 84; Phys. Rev. 130 (1963) 2529; 131 (1963) 2766. L. Mandel and E. Wolf, Rev. Mod. Phys. 37 (1965) 231. D. Stoler, Phys. Rev. Lett. 33 (1974) 1397. H.J. Carmichael and D.F. Walls, J. Phys. B9 (1976) 1199. H.J. Kimble and L. Mandel, Phys. Rev. AI3 (1976) 2123. H.J. Kimble and L. Mandel, Phys. Rev. AI5 (1977) 689.
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