- Email: [email protected]
Steady-state behavior of a V configuration three-level atom in a broadband squeezed vacuum bath
4 July 1994 PHYSICS LETTERS A
ELSEVIER
Physics Letters A 189 (1994) 449-453
Steady-state behavior of a V configuration three-level atom in a broadband squeezed vacuum bath Gao-xiang Li, Jin-sheng Peng CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China, National Laboratory of Laser Technology, Wuhan 430074, China, Department of Physics, Huazhong Normal University, Wuhan 430070, China
Received 10 February 1994; revised manuscript received 20 April 1994;accepted for publication 25 April 1994 Communicated by J.P. Vigier
Abstract
We have studied the steady-state behavior ofa V configuration three-level atom driven by a single-mode laser field in a broadband squeezed vacuum bath. The influences of the bath and the relative directions of the atomic dipole moments are discussed.
1. Introduction
The generation o f a squeezed state o f the radiation field [ 1 ] with reduced quantum fluctuations in one quadrature component has given new impetus to the studies o f atom-field interaction. Gardiner [ 2 ] has shown that one quadrature o f a two-level atomic dipole in a broadband squeezed vacuum decays with an enhanced rate and the other with a reduced rate compared to the normal atomic decay. Carmichael et al. [ 3 ] found that the spectral lines o f resonance fluorescence in the system o f a driven two-level atom interacting with a broadband squeezed vacuum can be narrowed to far below their natural linewidths due to the effects o f squeezed vacuum. The modification o f broadband squeezed vacuum on the quantum fluctuations of the optical forces exerted on lasercooled two-level atoms has been discussed by Shevy et al. [4]. They found that near the resonant region, the atom in a standing-wave field can reach a lower temperature than the Doppler limit. Palma and Knight [ 5 ] showed that the steady state o f a system including two two-level atoms or N two-level atoms
[6] interacting collectively with a broadband squeezed bath may evolve into a pure state which can be called atomic squeezed state [7 ]. Joshi and Purl [ 8 ] have discussed the steady-state behavior o f a driven three-level atom in cascade or A configuration interacting with two independent squeezed reservoirs. The results showed the steady-state populations of these two kinds of three-level atoms are nearly independent o f the parameters describing the squeezing o f the two reservoirs. In this paper, we investigate the steady-state behavior o f a V configuration three-level atom (Fig. 1 ) interacting with a single-mode laser field in a broadband squeezed vacuum bath. The results show that the steady-state populations are related to the parameters describing the squeezing o f the bath when the atomic dipole moment d2~ is perpendicular to d3~, but when the atomic dipole moments satisfy d2t = d3t, the steady-state behavior of the atom is independent of the parameters describing the squeezing of the bath, and coherent population trapping o f the two upper atomic states can occur.
0375-9601/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0375-9601 (94) 00332-J
G. Li, J. Peng / Physics Letters A 189 (1994) 449-453
450
A ---'~ . . . . . . . . . . . . . . .
The bandwidth of the squeezing field is broad enough to appear to the atom as a ~-correlated squeezed white noise. The correlation functions for the multimode squeezed field can then be written as [2-6,8]
la>
la)
( a +,o( t )ao~,( t') ) =n( 09)~( 09-09')tJ( t - t ' ) , ( a o A t ) a Z , ( t ' ) ) = [n(09) + 1 ]fi(09- 0 9 ' ) ~ ( t - t ' ) ,
O) L
( a,o( t )a,o, ( t') ) = M ( 09)~( 09+09'-- 209L)~( t--t') × exp[ --2i(09Lt+~L) ] ,
roll>
n=n(09)l,o=o~L,
Fig. 1. Diagram of the atom-field coupling system.
2. Model and master equation
We consider a three-level atom with two upper levels 13), 12) and the groundlevel I1 ) of energies 093, 092, 091 ( h = 1 ). The atom interacts with a single-mode cw laser of frequency COLand with the quantized multimode radiation field. The laser drives both atomic transitions which occur from 12 ) to I 1 ), and 13 ) to I 1 ). The atomic dipole m o m e n t can be described by
D+=d21A21+d3tA31,
D-=(D+)
+ .
(1)
Here di~ = ( i l d [ 1 ) represents the atomic dipole element for the l i ) - , l1 ) transition ( i = 2 or 3). The laser field is given by
(2)
where E is the unit polarization vector. The Hamiltonian of the system in the dipole approximation and the rotating approximation can be written in the form
H= ~ 09jAil + i=1
d09 09a ~, + a,o + VA-L
(4)
n is proportional to the number of photons in the squeezed vacuum bath and M is the parameter related to the degree of squeezing, n and M are restricted by the relation IMI 2 ~
=~21 e x p [
--i(09Lt+~L
)] ,
P3~ =P3~ exp[--i(09Lt+~gL) ] ,
(5)
the equations for the time evolution of atomic density elements can be easily obtained as follows, /~33 = -- 2 y 3 ( n +
1 )/033 -- i (~QT3/913 -~'~13P31 )
- ( n + 1 )(~3112P23 "~-72113P32) ,
g=Eo{~, exp[ --i(09Lt+ ~0L)] +C* exp [i(09Lt+ tpL) ]} ,
M=M(09)I~o=o~L=IMIe i~.
(6)
P22 = - 2y2(n+ 1 )P22 - i (~2T2/~12 -~12/~2t ) - ( n + 1 ) (72113,032 +73112P23) , /033 +P22 +/911 = 1 ,
(7) (8)
/~21 = iA/~21 -- [73n+72(2n+ 1 ) ]P21 --Y2113(n+ 1 )/731 + il2T2 (p22 --Pll ) + i~QT3P23 (9)
-- 2M(72121Pl 2 "at-72131/~13) ,
+(iig(09)(eo~'d21A21+e~o'd31A31)a~o-Fh.c. ) , o VA-L = e x p [ --i(09Lt + ~L) ] (I212A~2 "1-~r213A13) +h.c.
(3) Here ao~, + a,o are the creation and annihilation operators for the field mode with frequency 09 and polarization unit vector e,o, t2~=Eo~*-d~ ( i = 2, 3 ). We assume the radiation field to be a broadband squeezed vacuum state, centered around the frequencies COL.
flal - - i ( A - 0932)/~31 - [ y E n + y 3 ( 2 n + 1 ) ]/~31 - Y3112( n + 1 )/~21 + il2T3 (P33 - P l i ) + i121'2P32 - 2M(73121/~12 + Y3131/~13) , /~32 = -- i0932P32 -- (72 + 7 3 ) ( n +
(10) 1 )P32
-- 73112 ( n q- 1 ) (P33 q- P22 ) -]- i-Ql 2/)31 - iaQ~3/ffl 2 +273 ll2npll ,
where
(1 1 )
G. LL J. Peng /Physics Letters A 189 (1994) 449-453 /I = O)L - - ((-02 - - (-O1 ) ,
straightforward calculations, the steady-state populations p~t are obtained as
(,032 = 0,)3 - - (-02 ,
02 st St P33 :P22 ---- 402_1_72 ( 3 n + 1 - 2 M ) ( n + 1 ) '
7abcd= ( 3--~--'~c3)dab "dca ' tOca ' 2 , and we have used the abbreviation 7i= 7~m ( i = 2, 3). It is evident that 7, is the natural linewidth of the transition I i) --. I 1 ) and 7~d represents the decay rate of the interference between the dipole moments d~d and dab [ 9 ]. The amplitude of 7~bcdis related to the relative directions between dab and d~d. If dab'd¢d= 0 corresponding to no interference between dab and d~d, 7a~d=O. On the contrary, when dab is parallel to d~d, ?abcdis a real constant and there is strong interference between the two atomic dipole moments. Hence, we can discuss the influences of the parameters n and M which represent the properties of the squeezed vacuum bath, and the relative directions between the atomic dipole moments dz~ and d31 on the steady-state behavior of the atom.
3. Steady-state behavior of the V configuration threelevel atom
For simplicity we only consider the resonant case ( d = 0 , t032=0), and we also restrict ourselves to the cases in which both atomic dipole moments d2~ and dal are perpendicular or parallel, and the moduli of d2t and d31 are equal. For the case d21"d31=0, we choose them as follows, d31 = Ida, lex,
dE, = Id2t Icy.
(12)
Ifd3~ is parallel to dE,, we write them in the form
d3~ =d2t = ld3t ]ex.
(13)
The polarization unit of the laser field is chosen as E= (ex+ey)/v/2. So the parameters 0 , ( i = 2 , 3) relating to the atomic Rabi oscillating frequency satisfy 012 =0T2
=013
=0T3
=0.
451
(14)
First, we study the steady-state populations p~t (i = 1, 2, 3 ) when the atomic dipole moment d2~ is perpendicular to d31. In this case, the atomic decay rates are 72 ~--'73 = ~ 2 1 2 1 = 7 3 1 3 1 = 7 , 73112 = 7 2 1 1 3 = 7 2 1 3 1 = 7 3 1 2 1 ~ - 0 .
Starting from Eqs. ( 6 ) - ( 1 1 ) and after lengthy but
202-t- 72(n d- l ) ( 3 n + 1 - 2 M )
p~:~ = 402 + 72(3n+ l _ 2 M ) ( n +
l) .
(15)
Here we have chosen ~ = 0 or re. From Eq. ( 15 ) we see that p~ ( i = l, 2, 3) are obviously dependent on the parameters n and M. In the limit of a normal vacuum batch (n = 0, M = 0 ), Eq. ( 15 ) reduces 02
,t st P33----P22-- 4 0 2 + 7 2 ,
p~t _
2 0 2 + 72 402_t_72 •
(16)
These are the steady-state populations of the V configuration three-level a~om interacting with a singlemode laser field in a normal vacuum bath. When n # 0 and M = 0, which reflects the bath as a thermal reservoir, pit2 (pit3) is smaller than when n = 0 and M = 0, and p~t1 is larger. With increasing mean photon numbers of the thermal reservoir, both atomic populations in the two upper states decrease, but p~t increases. If the bath is in the broadband squeezed vacuum state, Eq. ( 15 ) shows that all the steady-state populations are not only dependent on the mean photon number of the bath but also on the degree and direction of squeezing of the bath. When M = - V / ~ - n + 1 ) which means the bath is in the pure squeezed vacuum state whose squeezing direction is ~ = n, the atomic populations in the two upper states are maxima. With increasing M, pit2 and pit3 decrease. When M = x / ~ n + 1 ) describing the bath in the pure squeezed vacuum state of ~ = 0, pit2 and p~t3 are minima. Joshi and Puri [ 8 ] found that the steady-state populations of a driven cascade or A configuration threelevel atom in the two independent squeezed vacuum reservoirs are nearly independent of the parameters which represent the squeezing of the two reservoirs. It is evident that there is a minor difference between Joshi and Puri's result and ours here. The cause leading to this minor difference is that the reservoirs are uncorrelated, the effects of the squeezing parameters of both reservoirs on the elements of atomic dipole moments are independent in Ref. [ 8 ]. But here we consider that in the decay of the V configuration three-level atom in the single squeezed bath, which
452
G. LL J. Peng / Physics Letters A 189 (1994) 449-453
can be regarded as two strong correlated baths, the influences of the squeezed parameter M on the elements of atomic dipole moments are correlated. Second, we discuss the case for which the atomic dipole moments d21 =d31. In this condition, there is strong interference between d21 and d3~. From Eqs. ( 6 ) - ( 1 1 ) , we obtain the steady-state populations Pi'~ as p ~ t3
~St
=P22
1
~'~"2 ,
p]tl = 0 .
(17)
The above equations show that the V configuration three-level atom driven by the laser field can exhibit atomic coherent population trapping in the two upper states [ l 0,1 1 ] even for damping of the squeezed vacuum bath. This coherent population trapping phenomenon is not related to the properties of the bath. This means that for fixed polarization of the laser field, the relative direction of the atomic dipole moments d21 and d3j plays an important role in the steady-state populations of the atom. It is the strong interference between d2t and d3~ that brings about this atomic population coherent trapping phenomenon. In order to realize this phenomenon, we inspect the steady-state off-diagonal elements of the atomic density matrix. Substituting Eq. ( 17 ) into Eqs. ( 9 ) - ( 1 1 ), we get Pit, =P[tl = 0 ,
P[~=P[t2=-½.
In conclusion, we have studied the steady behavior of the V configuration three-level atom driven by a single-mode laser field in a broadband squeezed vacuum bath. The steady-state populations of the atom are not only dependent on the photon number n of the bath but also related to the squeezing parameter M when the atomic dipole moment d2~ is perpendicular to d31. On the contrary, when d2~ =d3t, the steady behavior of the atom is not related to the properties of the bath, the atom can evolve into the coherent population trapping state (13) - 12) ) / v / 2 and then, atomic coherent population trapping in the two upper atomic levels occurs.
Acknowledgement One of the authors (J.S.P.) would like to thank Professor Abdus Salam, the International Atomic Energy Agency, and UNESCO for hospitality at the International Center for Theoretical Physics, Trieste. J.S. Peng also wishes to thank Professor E Persico for helpful discussions. This work is supported by the National Natural Science Foundation of China.
(18)
From Eqs. (17) and (18), we can see that the atom driven by the single-mode laser field in the squeezed vacuum bath evolves into the pure state I ~A(or) > =ei¢(I 3> -- 12> )/x/'~.
4. Conclusions
(19)
It is evident that [ ~A(0) ) is just the coherent population trapping state of the V configuration three-level atom system [ 10,11 ]. In this state, there is a strong cancellation between the atomic transitions [2) ~-~[ 1 ) and [3) ~ [ 1 ). This cancellation induces that the interactions between the atom and the radiation fields (laser field + bath field) are decoupled, and the atom is trapped in its two upper states. Thus, it is just the decoupling between the atom and the radiation fields that brings about the steady behavior of the atom independent of the properties of the bath.
References [ 1] L.A. Wu, H.J. Kimble,J.L. Hall and H. Wu, Phys. Rev. Lett. 57 (1986) 2520; K. Bergmanand H.A. Haus, Opt. Lett. 16 ( 1991 ) 153; H.J. Kimble, Phys. Rep. 219 (1992) 227. [2] C.W. Gardiner, Phys. Rev. Leu. 56 (1986) 1917. [3] H.J. Carmichael,A.S. Lane and D.E Walls,Phys. Rev. Lett. 58 (1987) 2539; J. Mod. Opt. 34 (1987) 821. [4] Y. Shevy,Phys. Rev. Lett. 64 (1990) 2539; R. Graham, D.F. Walls and W.P. Zhang, Phys. Rev. A 44 ( 1991 ) 7777; Y. Shevy, B. Crosignani and A. Yariv, Phys. Rev. A 46 (1992) 1421. [51G.M. Patina and P.L. Knight, Phys. Rev.A 39 (1989) 1962. [ 6 ] G.S. Agarwaland R.R. Purl, Phys. Rev. A 45 (1992) 3782. [ 7 ] S.M. Barnett and M.-A. Dupertuis, J. Opt. Soc. Am. B 4 (1987) 505; P. Zhou and J.S. Peng, Phys. Rev. A 44 ( 1991 ) 3331. [8] A. Joshi and R.R. Puri, Phys. Rev. A 45 (1992) 2025.
G. LL J. Peng / Physics Letters A 189 (1994) 449-453 [9] G.S. Agarwal, in: Springer tracts in modem physics, Vol. 70. Quantum statistical theories of spontaneous emission and their relation to other approaches, ed. G. Hohler (Springer, Berlin, 1974); P.W. Milonni, Phys. Rep. 25 (1976) 1.
453
[10] J.S. Peng, G.X. Li and P. Zhou, Phys. Rev. A 46 (1992) 1516; J.S. Peng and G.X. Li, Phys. Lett. A 176 ( 1993 ) 230. [ i l ] D.A. Cardimona, M.G. Raymer and C.R. Stroud Jr., J. Phys. B 15 (1982) 55; D.A. Cardimona, Phys. Rev. A 41 (1990) 5016.
ELSEVIER
Physics Letters A 189 (1994) 449-453
Steady-state behavior of a V configuration three-level atom in a broadband squeezed vacuum bath Gao-xiang Li, Jin-sheng Peng CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China, National Laboratory of Laser Technology, Wuhan 430074, China, Department of Physics, Huazhong Normal University, Wuhan 430070, China
Received 10 February 1994; revised manuscript received 20 April 1994;accepted for publication 25 April 1994 Communicated by J.P. Vigier
Abstract
We have studied the steady-state behavior ofa V configuration three-level atom driven by a single-mode laser field in a broadband squeezed vacuum bath. The influences of the bath and the relative directions of the atomic dipole moments are discussed.
1. Introduction
The generation o f a squeezed state o f the radiation field [ 1 ] with reduced quantum fluctuations in one quadrature component has given new impetus to the studies o f atom-field interaction. Gardiner [ 2 ] has shown that one quadrature o f a two-level atomic dipole in a broadband squeezed vacuum decays with an enhanced rate and the other with a reduced rate compared to the normal atomic decay. Carmichael et al. [ 3 ] found that the spectral lines o f resonance fluorescence in the system o f a driven two-level atom interacting with a broadband squeezed vacuum can be narrowed to far below their natural linewidths due to the effects o f squeezed vacuum. The modification o f broadband squeezed vacuum on the quantum fluctuations of the optical forces exerted on lasercooled two-level atoms has been discussed by Shevy et al. [4]. They found that near the resonant region, the atom in a standing-wave field can reach a lower temperature than the Doppler limit. Palma and Knight [ 5 ] showed that the steady state o f a system including two two-level atoms or N two-level atoms
[6] interacting collectively with a broadband squeezed bath may evolve into a pure state which can be called atomic squeezed state [7 ]. Joshi and Purl [ 8 ] have discussed the steady-state behavior o f a driven three-level atom in cascade or A configuration interacting with two independent squeezed reservoirs. The results showed the steady-state populations of these two kinds of three-level atoms are nearly independent o f the parameters describing the squeezing o f the two reservoirs. In this paper, we investigate the steady-state behavior o f a V configuration three-level atom (Fig. 1 ) interacting with a single-mode laser field in a broadband squeezed vacuum bath. The results show that the steady-state populations are related to the parameters describing the squeezing o f the bath when the atomic dipole moment d2~ is perpendicular to d3~, but when the atomic dipole moments satisfy d2t = d3t, the steady-state behavior of the atom is independent of the parameters describing the squeezing of the bath, and coherent population trapping o f the two upper atomic states can occur.
0375-9601/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0375-9601 (94) 00332-J
G. Li, J. Peng / Physics Letters A 189 (1994) 449-453
450
A ---'~ . . . . . . . . . . . . . . .
The bandwidth of the squeezing field is broad enough to appear to the atom as a ~-correlated squeezed white noise. The correlation functions for the multimode squeezed field can then be written as [2-6,8]
la>
la)
( a +,o( t )ao~,( t') ) =n( 09)~( 09-09')tJ( t - t ' ) , ( a o A t ) a Z , ( t ' ) ) = [n(09) + 1 ]fi(09- 0 9 ' ) ~ ( t - t ' ) ,
O) L
( a,o( t )a,o, ( t') ) = M ( 09)~( 09+09'-- 209L)~( t--t') × exp[ --2i(09Lt+~L) ] ,
roll>
n=n(09)l,o=o~L,
Fig. 1. Diagram of the atom-field coupling system.
2. Model and master equation
We consider a three-level atom with two upper levels 13), 12) and the groundlevel I1 ) of energies 093, 092, 091 ( h = 1 ). The atom interacts with a single-mode cw laser of frequency COLand with the quantized multimode radiation field. The laser drives both atomic transitions which occur from 12 ) to I 1 ), and 13 ) to I 1 ). The atomic dipole m o m e n t can be described by
D+=d21A21+d3tA31,
D-=(D+)
+ .
(1)
Here di~ = ( i l d [ 1 ) represents the atomic dipole element for the l i ) - , l1 ) transition ( i = 2 or 3). The laser field is given by
(2)
where E is the unit polarization vector. The Hamiltonian of the system in the dipole approximation and the rotating approximation can be written in the form
H= ~ 09jAil + i=1
d09 09a ~, + a,o + VA-L
(4)
n is proportional to the number of photons in the squeezed vacuum bath and M is the parameter related to the degree of squeezing, n and M are restricted by the relation IMI 2 ~
=~21 e x p [
--i(09Lt+~L
)] ,
P3~ =P3~ exp[--i(09Lt+~gL) ] ,
(5)
the equations for the time evolution of atomic density elements can be easily obtained as follows, /~33 = -- 2 y 3 ( n +
1 )/033 -- i (~QT3/913 -~'~13P31 )
- ( n + 1 )(~3112P23 "~-72113P32) ,
g=Eo{~, exp[ --i(09Lt+ ~0L)] +C* exp [i(09Lt+ tpL) ]} ,
M=M(09)I~o=o~L=IMIe i~.
(6)
P22 = - 2y2(n+ 1 )P22 - i (~2T2/~12 -~12/~2t ) - ( n + 1 ) (72113,032 +73112P23) , /033 +P22 +/911 = 1 ,
(7) (8)
/~21 = iA/~21 -- [73n+72(2n+ 1 ) ]P21 --Y2113(n+ 1 )/731 + il2T2 (p22 --Pll ) + i~QT3P23 (9)
-- 2M(72121Pl 2 "at-72131/~13) ,
+(iig(09)(eo~'d21A21+e~o'd31A31)a~o-Fh.c. ) , o VA-L = e x p [ --i(09Lt + ~L) ] (I212A~2 "1-~r213A13) +h.c.
(3) Here ao~, + a,o are the creation and annihilation operators for the field mode with frequency 09 and polarization unit vector e,o, t2~=Eo~*-d~ ( i = 2, 3 ). We assume the radiation field to be a broadband squeezed vacuum state, centered around the frequencies COL.
flal - - i ( A - 0932)/~31 - [ y E n + y 3 ( 2 n + 1 ) ]/~31 - Y3112( n + 1 )/~21 + il2T3 (P33 - P l i ) + i121'2P32 - 2M(73121/~12 + Y3131/~13) , /~32 = -- i0932P32 -- (72 + 7 3 ) ( n +
(10) 1 )P32
-- 73112 ( n q- 1 ) (P33 q- P22 ) -]- i-Ql 2/)31 - iaQ~3/ffl 2 +273 ll2npll ,
where
(1 1 )
G. LL J. Peng /Physics Letters A 189 (1994) 449-453 /I = O)L - - ((-02 - - (-O1 ) ,
straightforward calculations, the steady-state populations p~t are obtained as
(,032 = 0,)3 - - (-02 ,
02 st St P33 :P22 ---- 402_1_72 ( 3 n + 1 - 2 M ) ( n + 1 ) '
7abcd= ( 3--~--'~c3)dab "dca ' tOca ' 2 , and we have used the abbreviation 7i= 7~m ( i = 2, 3). It is evident that 7, is the natural linewidth of the transition I i) --. I 1 ) and 7~d represents the decay rate of the interference between the dipole moments d~d and dab [ 9 ]. The amplitude of 7~bcdis related to the relative directions between dab and d~d. If dab'd¢d= 0 corresponding to no interference between dab and d~d, 7a~d=O. On the contrary, when dab is parallel to d~d, ?abcdis a real constant and there is strong interference between the two atomic dipole moments. Hence, we can discuss the influences of the parameters n and M which represent the properties of the squeezed vacuum bath, and the relative directions between the atomic dipole moments dz~ and d31 on the steady-state behavior of the atom.
3. Steady-state behavior of the V configuration threelevel atom
For simplicity we only consider the resonant case ( d = 0 , t032=0), and we also restrict ourselves to the cases in which both atomic dipole moments d2~ and dal are perpendicular or parallel, and the moduli of d2t and d31 are equal. For the case d21"d31=0, we choose them as follows, d31 = Ida, lex,
dE, = Id2t Icy.
(12)
Ifd3~ is parallel to dE,, we write them in the form
d3~ =d2t = ld3t ]ex.
(13)
The polarization unit of the laser field is chosen as E= (ex+ey)/v/2. So the parameters 0 , ( i = 2 , 3) relating to the atomic Rabi oscillating frequency satisfy 012 =0T2
=013
=0T3
=0.
451
(14)
First, we study the steady-state populations p~t (i = 1, 2, 3 ) when the atomic dipole moment d2~ is perpendicular to d31. In this case, the atomic decay rates are 72 ~--'73 = ~ 2 1 2 1 = 7 3 1 3 1 = 7 , 73112 = 7 2 1 1 3 = 7 2 1 3 1 = 7 3 1 2 1 ~ - 0 .
Starting from Eqs. ( 6 ) - ( 1 1 ) and after lengthy but
202-t- 72(n d- l ) ( 3 n + 1 - 2 M )
p~:~ = 402 + 72(3n+ l _ 2 M ) ( n +
l) .
(15)
Here we have chosen ~ = 0 or re. From Eq. ( 15 ) we see that p~ ( i = l, 2, 3) are obviously dependent on the parameters n and M. In the limit of a normal vacuum batch (n = 0, M = 0 ), Eq. ( 15 ) reduces 02
,t st P33----P22-- 4 0 2 + 7 2 ,
p~t _
2 0 2 + 72 402_t_72 •
(16)
These are the steady-state populations of the V configuration three-level a~om interacting with a singlemode laser field in a normal vacuum bath. When n # 0 and M = 0, which reflects the bath as a thermal reservoir, pit2 (pit3) is smaller than when n = 0 and M = 0, and p~t1 is larger. With increasing mean photon numbers of the thermal reservoir, both atomic populations in the two upper states decrease, but p~t increases. If the bath is in the broadband squeezed vacuum state, Eq. ( 15 ) shows that all the steady-state populations are not only dependent on the mean photon number of the bath but also on the degree and direction of squeezing of the bath. When M = - V / ~ - n + 1 ) which means the bath is in the pure squeezed vacuum state whose squeezing direction is ~ = n, the atomic populations in the two upper states are maxima. With increasing M, pit2 and pit3 decrease. When M = x / ~ n + 1 ) describing the bath in the pure squeezed vacuum state of ~ = 0, pit2 and p~t3 are minima. Joshi and Puri [ 8 ] found that the steady-state populations of a driven cascade or A configuration threelevel atom in the two independent squeezed vacuum reservoirs are nearly independent of the parameters which represent the squeezing of the two reservoirs. It is evident that there is a minor difference between Joshi and Puri's result and ours here. The cause leading to this minor difference is that the reservoirs are uncorrelated, the effects of the squeezing parameters of both reservoirs on the elements of atomic dipole moments are independent in Ref. [ 8 ]. But here we consider that in the decay of the V configuration three-level atom in the single squeezed bath, which
452
G. LL J. Peng / Physics Letters A 189 (1994) 449-453
can be regarded as two strong correlated baths, the influences of the squeezed parameter M on the elements of atomic dipole moments are correlated. Second, we discuss the case for which the atomic dipole moments d21 =d31. In this condition, there is strong interference between d21 and d3~. From Eqs. ( 6 ) - ( 1 1 ) , we obtain the steady-state populations Pi'~ as p ~ t3
~St
=P22
1
~'~"2 ,
p]tl = 0 .
(17)
The above equations show that the V configuration three-level atom driven by the laser field can exhibit atomic coherent population trapping in the two upper states [ l 0,1 1 ] even for damping of the squeezed vacuum bath. This coherent population trapping phenomenon is not related to the properties of the bath. This means that for fixed polarization of the laser field, the relative direction of the atomic dipole moments d21 and d3j plays an important role in the steady-state populations of the atom. It is the strong interference between d2t and d3~ that brings about this atomic population coherent trapping phenomenon. In order to realize this phenomenon, we inspect the steady-state off-diagonal elements of the atomic density matrix. Substituting Eq. ( 17 ) into Eqs. ( 9 ) - ( 1 1 ), we get Pit, =P[tl = 0 ,
P[~=P[t2=-½.
In conclusion, we have studied the steady behavior of the V configuration three-level atom driven by a single-mode laser field in a broadband squeezed vacuum bath. The steady-state populations of the atom are not only dependent on the photon number n of the bath but also related to the squeezing parameter M when the atomic dipole moment d2~ is perpendicular to d31. On the contrary, when d2~ =d3t, the steady behavior of the atom is not related to the properties of the bath, the atom can evolve into the coherent population trapping state (13) - 12) ) / v / 2 and then, atomic coherent population trapping in the two upper atomic levels occurs.
Acknowledgement One of the authors (J.S.P.) would like to thank Professor Abdus Salam, the International Atomic Energy Agency, and UNESCO for hospitality at the International Center for Theoretical Physics, Trieste. J.S. Peng also wishes to thank Professor E Persico for helpful discussions. This work is supported by the National Natural Science Foundation of China.
(18)
From Eqs. (17) and (18), we can see that the atom driven by the single-mode laser field in the squeezed vacuum bath evolves into the pure state I ~A(or) > =ei¢(I 3> -- 12> )/x/'~.
4. Conclusions
(19)
It is evident that [ ~A(0) ) is just the coherent population trapping state of the V configuration three-level atom system [ 10,11 ]. In this state, there is a strong cancellation between the atomic transitions [2) ~-~[ 1 ) and [3) ~ [ 1 ). This cancellation induces that the interactions between the atom and the radiation fields (laser field + bath field) are decoupled, and the atom is trapped in its two upper states. Thus, it is just the decoupling between the atom and the radiation fields that brings about the steady behavior of the atom independent of the properties of the bath.
References [ 1] L.A. Wu, H.J. Kimble,J.L. Hall and H. Wu, Phys. Rev. Lett. 57 (1986) 2520; K. Bergmanand H.A. Haus, Opt. Lett. 16 ( 1991 ) 153; H.J. Kimble, Phys. Rep. 219 (1992) 227. [2] C.W. Gardiner, Phys. Rev. Leu. 56 (1986) 1917. [3] H.J. Carmichael,A.S. Lane and D.E Walls,Phys. Rev. Lett. 58 (1987) 2539; J. Mod. Opt. 34 (1987) 821. [4] Y. Shevy,Phys. Rev. Lett. 64 (1990) 2539; R. Graham, D.F. Walls and W.P. Zhang, Phys. Rev. A 44 ( 1991 ) 7777; Y. Shevy, B. Crosignani and A. Yariv, Phys. Rev. A 46 (1992) 1421. [51G.M. Patina and P.L. Knight, Phys. Rev.A 39 (1989) 1962. [ 6 ] G.S. Agarwaland R.R. Purl, Phys. Rev. A 45 (1992) 3782. [ 7 ] S.M. Barnett and M.-A. Dupertuis, J. Opt. Soc. Am. B 4 (1987) 505; P. Zhou and J.S. Peng, Phys. Rev. A 44 ( 1991 ) 3331. [8] A. Joshi and R.R. Puri, Phys. Rev. A 45 (1992) 2025.
G. LL J. Peng / Physics Letters A 189 (1994) 449-453 [9] G.S. Agarwal, in: Springer tracts in modem physics, Vol. 70. Quantum statistical theories of spontaneous emission and their relation to other approaches, ed. G. Hohler (Springer, Berlin, 1974); P.W. Milonni, Phys. Rep. 25 (1976) 1.
453
[10] J.S. Peng, G.X. Li and P. Zhou, Phys. Rev. A 46 (1992) 1516; J.S. Peng and G.X. Li, Phys. Lett. A 176 ( 1993 ) 230. [ i l ] D.A. Cardimona, M.G. Raymer and C.R. Stroud Jr., J. Phys. B 15 (1982) 55; D.A. Cardimona, Phys. Rev. A 41 (1990) 5016.