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Effects of the binomial photon distribution on the dynamics of two two-level atoms
Volume 72, number 5
OPTICS COMMUNICATIONS
1 August 1989
E F F E C T S OF T H E B I N O M I A L P H O T O N D I S T R I B U T I O N ON THE DYNAMICS OF TWO TWO-LEVEL ATOMS M.P. S H A R M A 1, D.A. C A R D I M O N A and A. G A V R I E L I D E S Air Force Weapons Laboratory, Quantum Optics Branch, AFWL/AROM, Kirtland AFB, NM 8 7117-6008, USA Received 4 April 1989
The phenomenon of collapse and revival of the atomic coherence of two two-level atoms interacting with a single mode quantized radiation field in the binomial state is studied theoretically. It is shown that the dynamics of the atomic system is quite sensitive to the photon statistics of the field.
1. Introduction Recent experiments [ 1 ] have shown that it is possible to study the interaction of a single two-level atom with a quantized radiation field. In these lightmatter interactions, the atomic dynamics are found to be very sensitive to the statistical properties o f the light. If the radiation field is in a pure n u m b e r state, the population inversion exhibits sinusoidal oscillations between the two levels. This behavior is the same as that o f a two-level atom interacting with a classical field. If the field is in a coherent state, collapses and revivals of these population oscillations have been predicted [ 2 ] and observed [ 3 ]. Recently Barnett and Knight [ 4 ] have considered the dynamical evolution o f m a n y two-level atoms interacting with a single mode radiation field and have reported some very interesting and important observations. The dynamics of two two-level atoms interacting with a quantized radiation field in a coherent state have been reported very recently [5]. A few years back binomial states o f the radiation field were introduced [ 6 ]. A binomial state is a linear combination of n u m b e r states with the coefficients chosen such that the photon distribution is binomial. These states interpolate between coherent states and number states. They share the properties
o f both and reduce to each in different limits thus enabling us to study the dynamics o f the system as the field statistics change from that of pure number states to that of coherent states. Very recently the binomial field distribution has been shown to produce collapse and revival phenomena when interacting with two- [ 7 ] and three-level systems [ 8 ]. In this work we present a study of two two-level atoms interacting with a single mode quantized radiation field in the binomial state. We show that the phenomena o f collapse and revival of the atomic inversion is sensitive to the photon density as well as to the photon statistics. We find that the asymmetry o f the photon distribution function has a very strong effect on the character of the collapse and revival phenomena.
2. Model hamiltonian We consider a system consisting of two identical two-level atoms interacting with a single mode quantized radiation field. We assume that the field is resonant with the atomic transition. In the interaction picture, the hamiltonian for the system in the rotating wave and dipole approximations is 2
H=ih
~ 2j(ca~ - c * a j ) ,
( 1)
j=l
Permanent address: Center for Advances Studies, Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131.
where c* and c are the creation and annihilation operators for the field, a~ and aj are the raising and
0 0 3 0 - 4 0 1 8 / 8 9 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
291
Volume 72, number 5
OPTICS COMMUNICATIONS
lowering operators for the atomic states, and 2j is the atom-field coupling constant for thejth atom. In our model the wave function for the system can be written in terms of the eigenstates l a, fl, n ) = Iaft) × In), where Ia ) and I/7) represent the states of the atom 1 and 2 respectively and In) denotes the number state of the field. The atom-field wave function I~') can be written as I ~u(t) ) =Aaan(t) Ia, a, n ) +Aabn+ l (t)la, b, n + 1 ) +Aban+ 1(t)[b, a, n + 1 ) +Abbn+2(t)Ib, b, n + 2 ) ,
(2) where A.nn(t ) (with a, fl=a, b) is the probability amplitude for finding atom 1 in state ra ) , atom 2 in state Ifl), and the photon field in state In). The states l a) and Ib ) represent the upper and lower states of each atom. In order to study the dynamics of the system, we need to calculate the probability amplitudes, which one gets by solving the Schfi$dinger equation ih O I~u(t))=Hl~(t)).
+21~1 Aabn+ I ([)
-22x/-~1
+2,~Abb~+2(t)
(4)
,
(5)
Abb.+2(t) ,
(6)
/i~.+:(t) =--X i N/~H~2 Aabn+ 1 (t) -22x/~2
Ab..+, (t) ,
(7)
for the probability amplitudes. Using the Laplace transform method and considering that both atoms are in the upper state initially with 21=22=2, we get
Aaa~(t)=A~(t) =2n+---~n+2+ ~ n + l 292
(9)
Here p = ~/N, where ~ is the average number of photons in the field and N is the maximum number of photons possible in the field (for more details on the binomial distribution see ref. [6] ). The total probability for finding both atoms in the upper state simultaneously is then given by P, (t) = ~ P. IA,(t) I2 n
= ~ P~ \ 2 n + 3 +
5c°sn+l -5
l-[ n + 2 X2
1 ( n + l )2
+
2 ( n + 2 ) ( n + 1) c o s ( ~ 2 0 (2n+3)2
1 ( n + l ) 2 cos(Zx//~-+62t)-].j + 2 (2n+3) 2
(10)
3. Discussion
Aaan(t)
Aban+i(t)=--~.l.~+ l Aaan(t) +,~2~
M
P n - ( N - n ) ! n! pn(1-p)N-n"
A,b,+ l (t)
Aba.+l(t) , =
as the solution for the probability amplitude for finding both atoms in the upper state simultaneously. We assume that the radiation field is in the binomial state with an initial photon distribution given by
(3)
Substituting the expressions for I ~u(t) ) and H from eqs. ( 1 ) and (2) into eq. (3) and applying the orthonormality property of the eigenfunctions, we get the following coupled equations,
/i~,(t) =22 n ~ l
1 August 1989
cos(~2t)
(8)
In this work we studied the effect of the initial mean photon number, ~, and the initial photon distribution on the probability of finding both atoms in the upper state simultaneously. Using eq. ( 11 ), we numerically computed P1 (t) as a function of time. In fig. 1, the ~'s that were used were small enough for the distribution to be symmetric (see fig. 3). Therefore, in fig. l increasing ri effectively increases the initial photon density. In fig. la, ~= 5 and PI (t) just begins to show collapses and revivals. As r~ is increased from l0 (fig. lb) to 24 (fig. lc), the collapses and revivals become more distinct and well separated. These three figs. (1 a-c) show the effect of the intensity (initial mean photon number) on the phenomena of collapse and revival. The shape of the photon distribution for these figures has not
o .
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.
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' 30.0
, 35.0
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' 45.0
, 50.0
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5 .0
60.0
t
0.0
5.0
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15.0
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2 .0
30.0
t Fig. I. P~ (t) with A= l for a binomial field distribution with (a) ~ = 5, p=0.1667, (b) if= lO, p=0.333, and (c) ff=24, p = 0 . 8 .
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15.0
i
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i
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i
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i
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i
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i
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0.0
5.0
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~5.0
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50.0
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45.0
50.0
55.0
60.0
t
0.0
5.0
10.0
15.0
20.0
25.0
30.0
t. Fig. 2. As in fig. 1 w i t h ( a ) f i = 2 7 , p = 0 . 9 , ( b ) ~ = 2 8 , p = 0 . 9 3 3 3 , a n d ( c ) ~ i = 3 0 , p = 0 . 9 9 9 9 .
Volume 72, number 5
OPTICS COMMUNICATIONS
1 August 1989
o
e: o ¸
n=30
o-
n=2 8 ¢t/
n=5
n=34
n=27/~
I
c5-
c~
~ ;
;
;
~ ;
; 1'o 1'1 l'z 13 A 1'~ 1'6 b 1'8 l'gz'o ~ 2'zz'3z'~ i~z'e z'~z'sz'93'o n
Fig. 3. The six binomial photon distributions used in figs. 1 and 2, showing the symmetry of the a=5, 10, and 24 distributions and the increasing asymmetry when a = 27, 28, and 30.
changed much (see fig. 3 ). In fig. 2, the distributions were asymmetric due to the proximity o f a to N (see fig. 3 ). Now, increasing a implies changing the shape of the photon distribution (essentially making it more narrow). On the basis o f fig. 1, when a = 2 7 (fig. 2a) we would have expected the collapses and revivals to be even more distinct and separated. Instead, we see a smearing of the phenomenon. As a increases from 28 (fig. 2b) to near 30 (fig. 2c), the smearing becomes more pronounced until finally all that survives are classical Rabi oscillations. In fig. 3 we plot the six binomial photon distributions used in figs. 1 and 2. By examining the photon distributions in this figure, we conclude that it is the asymmetry of the photon distribution, and not the increased intensity, which is responsible for the smearing effect on the p h e n o m e n a of collapse and revival o f the population oscillations. The trend in the dynamics described above defies intuition somewhat. The photon distributions used in fig. 1 were symmetric and therefore very similar in nature to the poissonian distributions o f coherent
states, which are usually considered to be the most classical quantized fields. However, these symmetric functions have a spread in photon numbers, and hence a spread in intensity, which is n o t a classical characteristic. It is this nonclassical nature that leads to the q u a n t u m collapses and revivals. To complete the peculiarity, when the photo distribution most resembles a n u m b e r state (as q u a n t u m mechanical as you can get), with very little uncertainty in the intensity (as in a classical field), the dynamical behavior o f the system consists purely o f classical Rabi oscillations.
References [ 1] D. Meschede, H. Walther and G. Muller, Phys. Rev. Lett. 54 (1985) 551; S. Haroche and J.M. Raimond, in: Advances in atomic and molecular physics, Vol. 20, eds. D. Bates and B. Bederson (Academic, N.Y. 1985) p. 347. [2] J.H. Eberly, N.B. Narezhny and J.J. Sanchez-Mondragon, Phys. Rev. Lett. 44 (1980) 1323; N.B. Narozhny, J.J. Sanchez-Mondragon and J.H. Eberly, Phys. Rev. 23 ( 1981 ) 236. 295
Volume 72, number 5
OPTICS COMMUNICATIONS
[3] G. Rempe, H. Walther and N. Klein, Phys. Rev. Lett. 58 (1987) 353. [4] S.M. Barnett and P.L. Knight, Optica Acta 31 (1984) 435. [5] Z. Deng, Optics Comm. 54 (1985) 222; M.S. Iqbal, S. Mahmood, M.S.K. Razmi and M.S. Zubairy, J. Opt. Soc. Am. B 5 (1988) 1312.
296
1 August 1989
[6] D. Stoler, B.E.A. Saleh and M.C. Teich, Optica Acta 32 (1985) 345. [7] A. Joshi and R.R. Purl, J. Mod. Optics 34 (1987) 1421. [8] M.E. Goggin, M.P. Sharma and A. Gavrielides, J. Mod. Optics, to be published.
OPTICS COMMUNICATIONS
1 August 1989
E F F E C T S OF T H E B I N O M I A L P H O T O N D I S T R I B U T I O N ON THE DYNAMICS OF TWO TWO-LEVEL ATOMS M.P. S H A R M A 1, D.A. C A R D I M O N A and A. G A V R I E L I D E S Air Force Weapons Laboratory, Quantum Optics Branch, AFWL/AROM, Kirtland AFB, NM 8 7117-6008, USA Received 4 April 1989
The phenomenon of collapse and revival of the atomic coherence of two two-level atoms interacting with a single mode quantized radiation field in the binomial state is studied theoretically. It is shown that the dynamics of the atomic system is quite sensitive to the photon statistics of the field.
1. Introduction Recent experiments [ 1 ] have shown that it is possible to study the interaction of a single two-level atom with a quantized radiation field. In these lightmatter interactions, the atomic dynamics are found to be very sensitive to the statistical properties o f the light. If the radiation field is in a pure n u m b e r state, the population inversion exhibits sinusoidal oscillations between the two levels. This behavior is the same as that o f a two-level atom interacting with a classical field. If the field is in a coherent state, collapses and revivals of these population oscillations have been predicted [ 2 ] and observed [ 3 ]. Recently Barnett and Knight [ 4 ] have considered the dynamical evolution o f m a n y two-level atoms interacting with a single mode radiation field and have reported some very interesting and important observations. The dynamics of two two-level atoms interacting with a quantized radiation field in a coherent state have been reported very recently [5]. A few years back binomial states o f the radiation field were introduced [ 6 ]. A binomial state is a linear combination of n u m b e r states with the coefficients chosen such that the photon distribution is binomial. These states interpolate between coherent states and number states. They share the properties
o f both and reduce to each in different limits thus enabling us to study the dynamics o f the system as the field statistics change from that of pure number states to that of coherent states. Very recently the binomial field distribution has been shown to produce collapse and revival phenomena when interacting with two- [ 7 ] and three-level systems [ 8 ]. In this work we present a study of two two-level atoms interacting with a single mode quantized radiation field in the binomial state. We show that the phenomena o f collapse and revival of the atomic inversion is sensitive to the photon density as well as to the photon statistics. We find that the asymmetry o f the photon distribution function has a very strong effect on the character of the collapse and revival phenomena.
2. Model hamiltonian We consider a system consisting of two identical two-level atoms interacting with a single mode quantized radiation field. We assume that the field is resonant with the atomic transition. In the interaction picture, the hamiltonian for the system in the rotating wave and dipole approximations is 2
H=ih
~ 2j(ca~ - c * a j ) ,
( 1)
j=l
Permanent address: Center for Advances Studies, Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131.
where c* and c are the creation and annihilation operators for the field, a~ and aj are the raising and
0 0 3 0 - 4 0 1 8 / 8 9 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
291
Volume 72, number 5
OPTICS COMMUNICATIONS
lowering operators for the atomic states, and 2j is the atom-field coupling constant for thejth atom. In our model the wave function for the system can be written in terms of the eigenstates l a, fl, n ) = Iaft) × In), where Ia ) and I/7) represent the states of the atom 1 and 2 respectively and In) denotes the number state of the field. The atom-field wave function I~') can be written as I ~u(t) ) =Aaan(t) Ia, a, n ) +Aabn+ l (t)la, b, n + 1 ) +Aban+ 1(t)[b, a, n + 1 ) +Abbn+2(t)Ib, b, n + 2 ) ,
(2) where A.nn(t ) (with a, fl=a, b) is the probability amplitude for finding atom 1 in state ra ) , atom 2 in state Ifl), and the photon field in state In). The states l a) and Ib ) represent the upper and lower states of each atom. In order to study the dynamics of the system, we need to calculate the probability amplitudes, which one gets by solving the Schfi$dinger equation ih O I~u(t))=Hl~(t)).
+21~1 Aabn+ I ([)
-22x/-~1
+2,~Abb~+2(t)
(4)
,
(5)
Abb.+2(t) ,
(6)
/i~.+:(t) =--X i N/~H~2 Aabn+ 1 (t) -22x/~2
Ab..+, (t) ,
(7)
for the probability amplitudes. Using the Laplace transform method and considering that both atoms are in the upper state initially with 21=22=2, we get
Aaa~(t)=A~(t) =2n+---~n+2+ ~ n + l 292
(9)
Here p = ~/N, where ~ is the average number of photons in the field and N is the maximum number of photons possible in the field (for more details on the binomial distribution see ref. [6] ). The total probability for finding both atoms in the upper state simultaneously is then given by P, (t) = ~ P. IA,(t) I2 n
= ~ P~ \ 2 n + 3 +
5c°sn+l -5
l-[ n + 2 X2
1 ( n + l )2
+
2 ( n + 2 ) ( n + 1) c o s ( ~ 2 0 (2n+3)2
1 ( n + l ) 2 cos(Zx//~-+62t)-].j + 2 (2n+3) 2
(10)
3. Discussion
Aaan(t)
Aban+i(t)=--~.l.~+ l Aaan(t) +,~2~
M
P n - ( N - n ) ! n! pn(1-p)N-n"
A,b,+ l (t)
Aba.+l(t) , =
as the solution for the probability amplitude for finding both atoms in the upper state simultaneously. We assume that the radiation field is in the binomial state with an initial photon distribution given by
(3)
Substituting the expressions for I ~u(t) ) and H from eqs. ( 1 ) and (2) into eq. (3) and applying the orthonormality property of the eigenfunctions, we get the following coupled equations,
/i~,(t) =22 n ~ l
1 August 1989
cos(~2t)
(8)
In this work we studied the effect of the initial mean photon number, ~, and the initial photon distribution on the probability of finding both atoms in the upper state simultaneously. Using eq. ( 11 ), we numerically computed P1 (t) as a function of time. In fig. 1, the ~'s that were used were small enough for the distribution to be symmetric (see fig. 3). Therefore, in fig. l increasing ri effectively increases the initial photon density. In fig. la, ~= 5 and PI (t) just begins to show collapses and revivals. As r~ is increased from l0 (fig. lb) to 24 (fig. lc), the collapses and revivals become more distinct and well separated. These three figs. (1 a-c) show the effect of the intensity (initial mean photon number) on the phenomena of collapse and revival. The shape of the photon distribution for these figures has not
o .
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.
0.0
5.0
.
.
10.0
.
15.0
20.0
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' 30.0
, 35.0
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' 45.0
, 50.0
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t Fig. I. P~ (t) with A= l for a binomial field distribution with (a) ~ = 5, p=0.1667, (b) if= lO, p=0.333, and (c) ff=24, p = 0 . 8 .
o.
o
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0.0
i
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|
10.0
i
15.0
i
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i
25.0
i
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5.0
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20.0
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50.0
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t
0.0
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10.0
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t. Fig. 2. As in fig. 1 w i t h ( a ) f i = 2 7 , p = 0 . 9 , ( b ) ~ = 2 8 , p = 0 . 9 3 3 3 , a n d ( c ) ~ i = 3 0 , p = 0 . 9 9 9 9 .
Volume 72, number 5
OPTICS COMMUNICATIONS
1 August 1989
o
e: o ¸
n=30
o-
n=2 8 ¢t/
n=5
n=34
n=27/~
I
c5-
c~
~ ;
;
;
~ ;
; 1'o 1'1 l'z 13 A 1'~ 1'6 b 1'8 l'gz'o ~ 2'zz'3z'~ i~z'e z'~z'sz'93'o n
Fig. 3. The six binomial photon distributions used in figs. 1 and 2, showing the symmetry of the a=5, 10, and 24 distributions and the increasing asymmetry when a = 27, 28, and 30.
changed much (see fig. 3 ). In fig. 2, the distributions were asymmetric due to the proximity o f a to N (see fig. 3 ). Now, increasing a implies changing the shape of the photon distribution (essentially making it more narrow). On the basis o f fig. 1, when a = 2 7 (fig. 2a) we would have expected the collapses and revivals to be even more distinct and separated. Instead, we see a smearing of the phenomenon. As a increases from 28 (fig. 2b) to near 30 (fig. 2c), the smearing becomes more pronounced until finally all that survives are classical Rabi oscillations. In fig. 3 we plot the six binomial photon distributions used in figs. 1 and 2. By examining the photon distributions in this figure, we conclude that it is the asymmetry of the photon distribution, and not the increased intensity, which is responsible for the smearing effect on the p h e n o m e n a of collapse and revival o f the population oscillations. The trend in the dynamics described above defies intuition somewhat. The photon distributions used in fig. 1 were symmetric and therefore very similar in nature to the poissonian distributions o f coherent
states, which are usually considered to be the most classical quantized fields. However, these symmetric functions have a spread in photon numbers, and hence a spread in intensity, which is n o t a classical characteristic. It is this nonclassical nature that leads to the q u a n t u m collapses and revivals. To complete the peculiarity, when the photo distribution most resembles a n u m b e r state (as q u a n t u m mechanical as you can get), with very little uncertainty in the intensity (as in a classical field), the dynamical behavior o f the system consists purely o f classical Rabi oscillations.
References [ 1] D. Meschede, H. Walther and G. Muller, Phys. Rev. Lett. 54 (1985) 551; S. Haroche and J.M. Raimond, in: Advances in atomic and molecular physics, Vol. 20, eds. D. Bates and B. Bederson (Academic, N.Y. 1985) p. 347. [2] J.H. Eberly, N.B. Narezhny and J.J. Sanchez-Mondragon, Phys. Rev. Lett. 44 (1980) 1323; N.B. Narozhny, J.J. Sanchez-Mondragon and J.H. Eberly, Phys. Rev. 23 ( 1981 ) 236. 295
Volume 72, number 5
OPTICS COMMUNICATIONS
[3] G. Rempe, H. Walther and N. Klein, Phys. Rev. Lett. 58 (1987) 353. [4] S.M. Barnett and P.L. Knight, Optica Acta 31 (1984) 435. [5] Z. Deng, Optics Comm. 54 (1985) 222; M.S. Iqbal, S. Mahmood, M.S.K. Razmi and M.S. Zubairy, J. Opt. Soc. Am. B 5 (1988) 1312.
296
1 August 1989
[6] D. Stoler, B.E.A. Saleh and M.C. Teich, Optica Acta 32 (1985) 345. [7] A. Joshi and R.R. Purl, J. Mod. Optics 34 (1987) 1421. [8] M.E. Goggin, M.P. Sharma and A. Gavrielides, J. Mod. Optics, to be published.