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Statistical properties of three-level atom resonance fluorescence
OPTICS COMMUNICATIONS
Volume 26, number 2
STATISTICAL
PROPERTIES OF THREE-LEVEL
August 1978
ATOM RESONANCE FLUORESCENCE
Bozena SOBOLEWSKA and Roman SOBOLEWSKI Institute of Physics, Polish Academy of Sciences, Al. Lotnikdw 32146, 02-668 Warszawa, Poland
Received 2 February 1978 Revised manuscript received 9 May 1978
Statistical properties of resonance fluorescence of a three-level atom, interacting with a monochromatic, strong laser field are examined. Analytical formulas, describing first- and second-order correlation functions are presented. The effect of antibunching is predicted and the nontrivial difference between the statistical properties of differently polarized fluorescence is discussed.
Statistical properties of the resonance fluorescence from a two-level quantum system have been recently studied both theoretically [1,2] and experimentally [3]. One of the most important features of the fluorescence light is the existence of negative photon correlation, called antibunching, which is a manifestation of the quantum nature of light. The occurence of antibunching was theoretically predicted not only in the resonance fluorescence, but also in some nonlinear processes [4-61. The experiment of Kimble et al. [3] is, until now, the only contkmation of this important statistical property of light. In this experiment as well as in all afore mentioned theoretical papers only the statistical properties of two-level atom resonance fluorescence were studied. There exist, however, various interesting physical effects, which involve the fluorescence of more complex quantum systems and cannot occur in the two-level case. One of such effects is the polarization dependence of various properties of the fluorescence light. The dependence of the spectral distribution of detected radiation on its polarization was recently pointed out and discussed [7-91. However, no research was done, until now, on the polarization dependence of the statistical properties of light. In this paper we present and discuss the analytical relations describing first- and second-order electric field correlation functions for two perpendicular polarization directions of the fluorescence from a three-level atom. The model of the atomic system under consideration is shown in fig. 1; exactly the same model was used in [8-lo]. The nondegenerate ground atomic state (J = 0) is coupled by the laser field E(f) = ixEo cos oLt to the mJ = 1 and m J = -1 sublevels of the excited state (J = 1). The laser beam is exactly at resonance with the atomic transition ,J = 0 *J = 1, m J = 0, which is strongly forbidden, Thus, we can neglect the existence of m J = 0 excited sublevel in further considerations. A typical experimental set-up, useful for the detection of the fluorescence light, is presented in fig. 2. An atom ic beam travels along the z-axis and a constant magnetic field B0 is applied in the same direction. The atoms are irradiated by a strong, single-mode laser beam of frequency wL, propagating alongy-direction and having a linear
m,=l
Fig. 1. The scheme of atomic levels. Energy difference be tween the ground state and the two interacting excited states is equal to fi(w~ + a) and h(w~ - a), respectively.
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OPTICS COMMUNICATIONS
Fig. 2. Experimental
August 1978
set-up.
polarization ix parallel to the x-axis. We can detect the fluorescence emitted along the z-axis with a linear polarization ix, or the fluorescence emitted along the x-axis with a linear polarization ir . The hamiltonian of the atomic system interacting with the classical field of the laser beam is taken in the form
&=#
+
J
++
;
;jcosULt,
(1)
where Q is the Zeeman splitting of the excited state, hu is a measure of the interaction energy. The Liouville-von Neumann equation for the atomic density matrix $(t) in our case can be written in the form d_ ih-@)=
^ ^
[~,P(Ol +&w.
(2)
The superoperator A describes radiative losses due to the spontaneous emission. Its matrix form contains only the decay constants I’ (I’ is the natural width of the excitedlevels). Components of the atomic dipole moment operator can be written as
&&,=d
[
The first- and second-order
;
;I
;
Dy=diy=d
correlation
functions
Cl’,’ (t, t + r) = U?;(t)_&i(t + 7)) a tiL(t)g;(t Gh&,(& t + 7, t + 7, t) = @;(f)_Qr
[_;
Ill]
;
-j
.
of the fluorescence
(3)
light can be expressed in the form:
+ 7)) ,
t r).!?$t + r)iJt)
(4a) a (s^t(t)$(t
+ r)i;(f
+ 7)$;(t))
,
(4b)
where p = x, y denotes the polarization direction of detected radiation; _!?;, f$ are positive and negative frequency parts of the p-component of electric field operator; ,!?c, $ are positive and negative frequency parts of the operator ifi. To evaluate these two-time correlation functions it is convenient to use the quantum “fluctuation-regression theorem” [12,13]. According to this theorem, the two-time average @US,> satisfies the same equation of motion as the one-time average G;(t)). The equations for (Si), or equivalently, the Bloch equations for the elements of the atomic density matrix fi can be solved by the method presented in [lo]. Besides the dipole and rotating wave approximation, the only other approximation made in the calculations is based on the assumption that 212
Volume 26, number 2
August 1978
OPTICS COMMUNICATIONS
the natural width of the excited levels (r) is much smaller than u. This means that the laser field is strong enough. We have found, in the first-order approximation with respect to r/u (“secular approximation”), the time evolu(2) (t, t + T, t + T, t). Detailed calculation of S:(t)) and the two-time correlation functions GLy(t, t + T) and G,,,, tions will be published elsewhere. The nontrivial difference between the statistical properties of differently polarized radiation can be readily seen by comparing the first-order correlation functions G$..(t, t + T) and G$(t, t t 7). G(L) a ~ xx M(tiu)2
[2u2e-7e7
t a2(e-
71 r+ e--Y2r) cos(6wr) t 2u 2e-
737
cos(26wr)l
eeiwLT ,
(54
xcos(bw) +-
(5b)
where M =
,
+ u2Q2 u2tfP (6w)2
6w = (2u2 + Cl.2
=r
’
’
y2 = r
6u4 t 9uQP t a4 2(6w)4
’
y3 = r
3u4 t 362 Sl2 + 524 (6 N4
’ 74= (So)4
We have assumed here that t 9 I’-l, so that our system is already in the steady-state. For the Zeeman splitting a equal to zero G::)(t, t + T) - 0 (as well as all other correlation functions describing the y-polarized light). The formula for G$:(t, t + T) agrees then with the result obtained in [l]. For long time intervals r , G$)(t, t t T) tends to zero - all correlation disappear, while G::)(t, t + T) factorizes and becomes proportional to LS$t),ls,-(t + T)). This result means that there exists a coherent component of y-polarized radiation which does not vanish even for long time intervals r. The effect is caused by the existence of nonvanishing mean value of i$ : 2fiu(iJ2 - SP) M
C!$(t))z+i
(6)
which plays the role of a classical source of radiation. g;(t)) vanishes only for s2 = 0 and a = u. C? = u is characteristic point for our system. It corresponds to equal intensities of x-polarized and y-polarized light. For the x-polarized fluorescence C!?:(t)) t0, so in this case there is no coherent component of radiation. With the help of the quantum “fluctuation-regression theorem” [12,13] it is easy to show that the second order correlation function GC2) pp,,,,(t, t + 7, t + 7, t) can be expressed as a product of two first-order correlation functions (as in the two-level case [2]): G;fp,(t,
t + T, t + T, t) = G$(t,
One-time correlation
G$(t
functions
>t) -?!f.!_e--Ysf a
G$
t)G$(q
T) .
for arbitrary
(7)
t are given by:
cos(26wt) -
,
(84
(fjN2
2uQP (49 _ a2)($ e--7zt cos(bwt) t ____ e-Tat cos(26 wt) t 2u2 M(6 w)~ (6 N4 (6 44 We define the normalized intensity correlation function NJ as G$(t,
t) a
_
N’ f [G;$Jt,
8u2522
t + T, t + T, t) - G,$(t, t)G$(t
+ T, t + r)]/Gi;(t,
t)G$(t
t T, t + T)
_ Q2)e-74t+Fe (8b) (9)
213
W is a measure
August 1978
OPTICS COMMUNICATIONS
Volume 26, number 2
of the light fluctuations.
Then, from eq. (7) we obtain:
NJ = G,$)(T, ~)/Gi;(t t T, t t T) - 1 . Rutting t% I’-l xx=__
(10)
we have:
IV
e--7V cos(2bwr) -
2IJ2(&.# AYE-_ __2M (W4
fi2(u2- a212
e--Yz7 COS@WT)t -
(1la)
e--y47
2u2(6w)2 M
2@w)4
e--y3*
cos(2&4+(4~2-
’ Q2)(u2-
2(6w)4
Q2)e-y47
(1lb)
The following conclusions can be drawn from these results: 1) For r + 0, Ax +V” + -1 for each value of s1. Thus small, time intervals T correspond to antibunching in the fluorescence light. This property is true for all values oft, not only for the stationary state. 2) For a = 0 the expression for AX agrees with the results of [l] and there exists a well defined limit of AJ’. 3) When Cl increases, the Rabi mutation frequency 26~ increases and maxima of Xx increase, too. Maxima of V’ have the smallest values for s2 close to u, they increase with increasing a and reach the saturation for a -+ 00. 4) With increasing r the amplitudes of oscillations decrease and for r -+ 00, XX and V’ attain the steady-state values equal to zero. This means that all correlations disappear. The above described properties are illustrated in figs. 3a and 3b where the r-dependence of Ax and hJ’ for some fixed values of s2/u is shown,
(b)
x,??
n
Fig. 3. (a) The normalized intensity correlation functions hX, 0’ for various values of the Zeeman splitting fi and the ratio r/u = 0.1 (t % r-l). (b) The normalized intensity correlation functions hX, ti for n/u = 3 and I’/u = 0.1 (? B- I+-‘).
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Volume 26, number 2
OPTICS COMMUNICATIONS
August 1978
One can see, that as in the case of a two-level atom, our system passes periodically from the state of negative correlations to the state of positive correlations. For the y-polarized radiation these changes are twice slower than for the x-polarized light - the states of bunching and antibunching last longer. It is also evident that the joint probability of detecting the second photon at time t + T, while the first photon was detected at time moment t, decreases more slowly with increasing fi than the total intensity of radiation at time t. This is the cause of the growth of AMwith increasing SL The fact that the fluctuations of the y-polarized fluorescence are smaller than those of the x-polarized radiation can be interpreted as due, among others, to the existence of the coherent component in the y-polarized light, for which X = 0. We examined the normalized correlation functions AMnot only in the steady-state (t % r-l), but also for short times, t corresponding to the transient regime. We have found that for t equal, for instance, to the period of G,($(t, t) the effect of antibunching dominates, in the sense that Ax and Au are negative for almost all time inter-, vals (see fig. 4). The authors wish to thank Professor Z. Bialynicka-Birula for many helpful suggestions, Dr. S.J. Lewandowski for careful reading the text and Dr. Z. Polch for numerical computations.
References [l] [2] [3] [4] [S] [6] [7] [ 81 [9] [IO] [ll] [ 121 [13]
H.J. Carmichael and D.F. Walls, J. Phys. B9 (1976) 1199. H.J. Kimble and L. Mandel, Phys. Rev. Al3 (1976) 2123. J.H. Kimble, M. Dagenais and L. Mandel, Phys. Rev. Lett. 39 (1977) 691. D. Stoler, Phys. Rev. Lett. 33 (1974) 1397. M. Kozierowski and R. TanaS, Opt. Commun. 21 (1977) 229. J. Mostowski and K. Rzazewski, to be published. C. Cohen-Tannoudji and S. Reynaud, J. Phys. BlO (1977) 2311. B. Sobolewska, Ph.D. Thesis (Institute of Physics, Polish Academy of Sciences), unpublished. R. Kornblith and J.H. Eberly, to be published. B. Sobolowska, Opt. Commun. 19 (1976) 185. R.J. Glauber, Phys. Rev. 131 (1963) 2766. M. Lax, Phys. Rev. 172 (1968) 350. B.R. Mollow, Phys. Rev. Al2 (1975) 1919.
215
Volume 26, number 2
STATISTICAL
PROPERTIES OF THREE-LEVEL
August 1978
ATOM RESONANCE FLUORESCENCE
Bozena SOBOLEWSKA and Roman SOBOLEWSKI Institute of Physics, Polish Academy of Sciences, Al. Lotnikdw 32146, 02-668 Warszawa, Poland
Received 2 February 1978 Revised manuscript received 9 May 1978
Statistical properties of resonance fluorescence of a three-level atom, interacting with a monochromatic, strong laser field are examined. Analytical formulas, describing first- and second-order correlation functions are presented. The effect of antibunching is predicted and the nontrivial difference between the statistical properties of differently polarized fluorescence is discussed.
Statistical properties of the resonance fluorescence from a two-level quantum system have been recently studied both theoretically [1,2] and experimentally [3]. One of the most important features of the fluorescence light is the existence of negative photon correlation, called antibunching, which is a manifestation of the quantum nature of light. The occurence of antibunching was theoretically predicted not only in the resonance fluorescence, but also in some nonlinear processes [4-61. The experiment of Kimble et al. [3] is, until now, the only contkmation of this important statistical property of light. In this experiment as well as in all afore mentioned theoretical papers only the statistical properties of two-level atom resonance fluorescence were studied. There exist, however, various interesting physical effects, which involve the fluorescence of more complex quantum systems and cannot occur in the two-level case. One of such effects is the polarization dependence of various properties of the fluorescence light. The dependence of the spectral distribution of detected radiation on its polarization was recently pointed out and discussed [7-91. However, no research was done, until now, on the polarization dependence of the statistical properties of light. In this paper we present and discuss the analytical relations describing first- and second-order electric field correlation functions for two perpendicular polarization directions of the fluorescence from a three-level atom. The model of the atomic system under consideration is shown in fig. 1; exactly the same model was used in [8-lo]. The nondegenerate ground atomic state (J = 0) is coupled by the laser field E(f) = ixEo cos oLt to the mJ = 1 and m J = -1 sublevels of the excited state (J = 1). The laser beam is exactly at resonance with the atomic transition ,J = 0 *J = 1, m J = 0, which is strongly forbidden, Thus, we can neglect the existence of m J = 0 excited sublevel in further considerations. A typical experimental set-up, useful for the detection of the fluorescence light, is presented in fig. 2. An atom ic beam travels along the z-axis and a constant magnetic field B0 is applied in the same direction. The atoms are irradiated by a strong, single-mode laser beam of frequency wL, propagating alongy-direction and having a linear
m,=l
Fig. 1. The scheme of atomic levels. Energy difference be tween the ground state and the two interacting excited states is equal to fi(w~ + a) and h(w~ - a), respectively.
211
Volume 26, number 2
OPTICS COMMUNICATIONS
Fig. 2. Experimental
August 1978
set-up.
polarization ix parallel to the x-axis. We can detect the fluorescence emitted along the z-axis with a linear polarization ix, or the fluorescence emitted along the x-axis with a linear polarization ir . The hamiltonian of the atomic system interacting with the classical field of the laser beam is taken in the form
&=#
+
J
++
;
;jcosULt,
(1)
where Q is the Zeeman splitting of the excited state, hu is a measure of the interaction energy. The Liouville-von Neumann equation for the atomic density matrix $(t) in our case can be written in the form d_ ih-@)=
^ ^
[~,P(Ol +&w.
(2)
The superoperator A describes radiative losses due to the spontaneous emission. Its matrix form contains only the decay constants I’ (I’ is the natural width of the excitedlevels). Components of the atomic dipole moment operator can be written as
&&,=d
[
The first- and second-order
;
;I
;
Dy=diy=d
correlation
functions
Cl’,’ (t, t + r) = U?;(t)_&i(t + 7)) a tiL(t)g;(t Gh&,(& t + 7, t + 7, t) = @;(f)_Qr
[_;
Ill]
;
-j
.
of the fluorescence
(3)
light can be expressed in the form:
+ 7)) ,
t r).!?$t + r)iJt)
(4a) a (s^t(t)$(t
+ r)i;(f
+ 7)$;(t))
,
(4b)
where p = x, y denotes the polarization direction of detected radiation; _!?;, f$ are positive and negative frequency parts of the p-component of electric field operator; ,!?c, $ are positive and negative frequency parts of the operator ifi. To evaluate these two-time correlation functions it is convenient to use the quantum “fluctuation-regression theorem” [12,13]. According to this theorem, the two-time average @US,> satisfies the same equation of motion as the one-time average G;(t)). The equations for (Si), or equivalently, the Bloch equations for the elements of the atomic density matrix fi can be solved by the method presented in [lo]. Besides the dipole and rotating wave approximation, the only other approximation made in the calculations is based on the assumption that 212
Volume 26, number 2
August 1978
OPTICS COMMUNICATIONS
the natural width of the excited levels (r) is much smaller than u. This means that the laser field is strong enough. We have found, in the first-order approximation with respect to r/u (“secular approximation”), the time evolu(2) (t, t + T, t + T, t). Detailed calculation of S:(t)) and the two-time correlation functions GLy(t, t + T) and G,,,, tions will be published elsewhere. The nontrivial difference between the statistical properties of differently polarized radiation can be readily seen by comparing the first-order correlation functions G$..(t, t + T) and G$(t, t t 7). G(L) a ~ xx M(tiu)2
[2u2e-7e7
t a2(e-
71 r+ e--Y2r) cos(6wr) t 2u 2e-
737
cos(26wr)l
eeiwLT ,
(54
xcos(bw) +-
(5b)
where M =
,
+ u2Q2 u2tfP (6w)2
6w = (2u2 + Cl.2
=r
’
’
y2 = r
6u4 t 9uQP t a4 2(6w)4
’
y3 = r
3u4 t 362 Sl2 + 524 (6 N4
’ 74= (So)4
We have assumed here that t 9 I’-l, so that our system is already in the steady-state. For the Zeeman splitting a equal to zero G::)(t, t + T) - 0 (as well as all other correlation functions describing the y-polarized light). The formula for G$:(t, t + T) agrees then with the result obtained in [l]. For long time intervals r , G$)(t, t t T) tends to zero - all correlation disappear, while G::)(t, t + T) factorizes and becomes proportional to LS$t),ls,-(t + T)). This result means that there exists a coherent component of y-polarized radiation which does not vanish even for long time intervals r. The effect is caused by the existence of nonvanishing mean value of i$ : 2fiu(iJ2 - SP) M
C!$(t))z+i
(6)
which plays the role of a classical source of radiation. g;(t)) vanishes only for s2 = 0 and a = u. C? = u is characteristic point for our system. It corresponds to equal intensities of x-polarized and y-polarized light. For the x-polarized fluorescence C!?:(t)) t0, so in this case there is no coherent component of radiation. With the help of the quantum “fluctuation-regression theorem” [12,13] it is easy to show that the second order correlation function GC2) pp,,,,(t, t + 7, t + 7, t) can be expressed as a product of two first-order correlation functions (as in the two-level case [2]): G;fp,(t,
t + T, t + T, t) = G$(t,
One-time correlation
G$(t
functions
>t) -?!f.!_e--Ysf a
G$
t)G$(q
T) .
for arbitrary
(7)
t are given by:
cos(26wt) -
,
(84
(fjN2
2uQP (49 _ a2)($ e--7zt cos(bwt) t ____ e-Tat cos(26 wt) t 2u2 M(6 w)~ (6 N4 (6 44 We define the normalized intensity correlation function NJ as G$(t,
t) a
_
N’ f [G;$Jt,
8u2522
t + T, t + T, t) - G,$(t, t)G$(t
+ T, t + r)]/Gi;(t,
t)G$(t
t T, t + T)
_ Q2)e-74t+Fe (8b) (9)
213
W is a measure
August 1978
OPTICS COMMUNICATIONS
Volume 26, number 2
of the light fluctuations.
Then, from eq. (7) we obtain:
NJ = G,$)(T, ~)/Gi;(t t T, t t T) - 1 . Rutting t% I’-l xx=__
(10)
we have:
IV
e--7V cos(2bwr) -
2IJ2(&.# AYE-_ __2M (W4
fi2(u2- a212
e--Yz7 COS@WT)t -
(1la)
e--y47
2u2(6w)2 M
2@w)4
e--y3*
cos(2&4+(4~2-
’ Q2)(u2-
2(6w)4
Q2)e-y47
(1lb)
The following conclusions can be drawn from these results: 1) For r + 0, Ax +V” + -1 for each value of s1. Thus small, time intervals T correspond to antibunching in the fluorescence light. This property is true for all values oft, not only for the stationary state. 2) For a = 0 the expression for AX agrees with the results of [l] and there exists a well defined limit of AJ’. 3) When Cl increases, the Rabi mutation frequency 26~ increases and maxima of Xx increase, too. Maxima of V’ have the smallest values for s2 close to u, they increase with increasing a and reach the saturation for a -+ 00. 4) With increasing r the amplitudes of oscillations decrease and for r -+ 00, XX and V’ attain the steady-state values equal to zero. This means that all correlations disappear. The above described properties are illustrated in figs. 3a and 3b where the r-dependence of Ax and hJ’ for some fixed values of s2/u is shown,
(b)
x,??
n
Fig. 3. (a) The normalized intensity correlation functions hX, 0’ for various values of the Zeeman splitting fi and the ratio r/u = 0.1 (t % r-l). (b) The normalized intensity correlation functions hX, ti for n/u = 3 and I’/u = 0.1 (? B- I+-‘).
214
Volume 26, number 2
OPTICS COMMUNICATIONS
August 1978
One can see, that as in the case of a two-level atom, our system passes periodically from the state of negative correlations to the state of positive correlations. For the y-polarized radiation these changes are twice slower than for the x-polarized light - the states of bunching and antibunching last longer. It is also evident that the joint probability of detecting the second photon at time t + T, while the first photon was detected at time moment t, decreases more slowly with increasing fi than the total intensity of radiation at time t. This is the cause of the growth of AMwith increasing SL The fact that the fluctuations of the y-polarized fluorescence are smaller than those of the x-polarized radiation can be interpreted as due, among others, to the existence of the coherent component in the y-polarized light, for which X = 0. We examined the normalized correlation functions AMnot only in the steady-state (t % r-l), but also for short times, t corresponding to the transient regime. We have found that for t equal, for instance, to the period of G,($(t, t) the effect of antibunching dominates, in the sense that Ax and Au are negative for almost all time inter-, vals (see fig. 4). The authors wish to thank Professor Z. Bialynicka-Birula for many helpful suggestions, Dr. S.J. Lewandowski for careful reading the text and Dr. Z. Polch for numerical computations.
References [l] [2] [3] [4] [S] [6] [7] [ 81 [9] [IO] [ll] [ 121 [13]
H.J. Carmichael and D.F. Walls, J. Phys. B9 (1976) 1199. H.J. Kimble and L. Mandel, Phys. Rev. Al3 (1976) 2123. J.H. Kimble, M. Dagenais and L. Mandel, Phys. Rev. Lett. 39 (1977) 691. D. Stoler, Phys. Rev. Lett. 33 (1974) 1397. M. Kozierowski and R. TanaS, Opt. Commun. 21 (1977) 229. J. Mostowski and K. Rzazewski, to be published. C. Cohen-Tannoudji and S. Reynaud, J. Phys. BlO (1977) 2311. B. Sobolewska, Ph.D. Thesis (Institute of Physics, Polish Academy of Sciences), unpublished. R. Kornblith and J.H. Eberly, to be published. B. Sobolowska, Opt. Commun. 19 (1976) 185. R.J. Glauber, Phys. Rev. 131 (1963) 2766. M. Lax, Phys. Rev. 172 (1968) 350. B.R. Mollow, Phys. Rev. Al2 (1975) 1919.
215