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Effect of dispersion forces on the dynamical evolution of the two-atom Dicke model in a broadband squeezed bath
Optics Communications North-Holland
94 ( 1992) 362-368
OPTICS COMMUNICATIONS
Effect of dispersion forces on the dynamical evolution of the two-atom Dicke model in a broadband squeezed bath Amitabh
Joshi and R.R. Puri
MDRS and Theoretical Physics Division, Bhabha Atomic Research Centre, Trombay, Bombay 400 085. India Received
13 February
1992; revised manuscript
received
7 July 1992
We show that the two-atom Dicke model, including dipole-dipole interaction (dispersion forces) and interaction with a broadband squeezed bath, exhibits a periodic modulation phenomenon in its population decay. For this purpose we have calculated explicitly the analytic expression for the time evolution of the density matrix elements under the intense field limit. The phase sensitive nature of the modulation phenomenon has been clearly brought out in these calculations.
1. Introduction The interaction of a system of atoms with a broadband squeezed bath of radiation having modes correlated in pairs gives rise to some unusual dynamic and steady state characteristics [ l-5 1. The most striking example is the case of a strongly driven two-level atom interacting with such a squeezed bath resulting in the narrowing of the central peak of the Mollow triplet or an equality of heights of all the three peaks [ 21. The steady state of a system of two-level atoms collectively interacting with such a squeezed bath can be a pure state under certain conditions [ 3-5 ] whereas the mixed state is attained under the influence of an ordinary bath. Thus these unusual properties are a manifestation of the correlations present in the squeezed bath. Here, we explore further the effects of a squeezed bath on the dynamical properties of a system of two-level atoms. The Dicke model of two two-level atoms interacting collectively with the squeezed bath and driven by an external field is considered here. Our aim is to explore dynamic behaviour of such a system in the high field limit. In our analysis the dipole-dipole interaction (dispersion forces) term in the master equation is included. This term is important in the Dicke model which assumes close spacing between the atoms ( -A). This term originates by retaining the atomic position dependent terms in the hamiltonian and going to the limit of small systems after eliminating bath operators from the equation for the density matrix [ 61. On the contrary, if, small system limit is assumed in writing the atom-field hamiltonian itself, such a term does not arise. In sec. 2, we introduce the model and master equation for the Dicke system of N identical two-level atoms interacting with a broadband squeezed bath. We then calculate the eigenvalues and eigenstates of the two-atom system hamiltonian. In sec. 3, we derive analytic expression for the evolution of the density matrix elements in the high field limit which are useful to obtain expectation values of the dynamic variables. This is followed by discussions and concluding remarks.
2. The model We consider a system of N identical two-level atoms each of transition frequency w. interacting collectively with a broadband squeezed bath which is pumped by a coherent field of frequency wL. The atom system is also driven by an external field, which is also of frequency wi_. Further, we assume the bandwidth of the squeezed 362
0030-4018/92/$
05.00 0 1992 Elsevier Science Publishers
B.V. All rights reserved.
1 December 1992
OPTICS COMMUNICATIONS
Volume 94, number 5
bath to be broader than the spectral width of the fluorescence so that the spectrum falls well within the bandwidths of the squeezed vacuum modes. The system can then be described by the following master equation [ 731 for the reduced density matrix (in Born-Markov approximation and in a frame rotating with wL): $/at=-id[S,,p]-i[oS++o!*S_,p]-i +n(2S+pS_
1 Q,[S.?S;,p]+y[(ti+1)(2S_pS+-S+S_p-pS+S_) i#J
-s_S+p--pS_S+)+M(2S+pS+
-s:p-ps:)+w(2s_ps_
-sz_p-pS2)]
)
(2.1)
where, A= coo- co=, M= 1MI exp (iv) and ST, S, are collective atomic operators and (Y= I a I exp ( igL) is the Rabi frequency associated with the field. Here, Q, (Q,=Q,,) is the coefficient of dipole-dipole interaction. Also, IMI 2< A( A+ 1)) the equality sign holds for an ideal squeezed bath. We consider here the ideal squeezed bath conditions. The master equation is S2 conserving, provided the dipole-dipole interaction term Qij= 0. The dipole-dipole interaction term does not, in general, conserve S* except for a special symmetric arrangement of the atoms [ 9 ] or for N= 2 in which case
(2.2) where VEQ,~=Q~,.
The master equation
then reads
ap~at=-i[~,,p]+~[(~+i)(2S_pS+-S+S_p-pS+S_)+n(2S+pS_-S_S+p-pS_S+) +M(2S+pS+
-s:p-ps:)+M*(2s_ps_
-SLp-pS2_)]
)
(2.3)
and &=A&+
Ial (S, +S_)+vSS.
(2.4)
Here we have redefined Sk exp ( f igL) as Sk and M= IMI exp (i@) with $= q- 2eL. Note that the hamiltonian Ho is known as the two atom Dicke model hamiltonian with dipole-dipole interaction taken into account [ lo]. The only states taking part in the dynamics are the triplet states ( ]e) , 1g) describe the excited and ground states of the two identical atoms, respectively) (2.5a) and the singlet state (2Sb) is completely decoupled optically from the triplet state. This would be understood from eq. (2.4) which is invariant under the exchange a c* b which implies that the symmetry of the atomic state is a constant of motion. The singlet state is antisymmetric while the triplet states are symmetric, which means that they cannot be coupled by the interaction with the field. In this case it is quite simple to calculate the new frequencies of the system by diagonalization of the effective hamiltonian Ho. If A # 0, the secular equation leading to the dressed eigenfrequencies is a cubic equation that can be best solved by numerical techniques. If A=O, instead, the secular equation is still a cubic equation, which can be best solved analytically and the eigenfrequencies of the hamiltonian (2.4) are given explicitly by A,=v,
3L2,3=v/2?K/2
and the corresponding
andK=(v*+16]~1*)“*, eigenstates
(2.6)
are 363
Iv2)=D;1[2Jzlal
lr~)=~(l-)-I1w>
)W+~$[2&t
1 December 1992
OPTICS COMMUNICATIONS
Volume 94, number 5
I-1)-(K-v)10)+2fiIaI
II>1 1
I-1>+(K-w)+2JZlal ll)],
D,,z=[161a!12+(K~~)211’2.
(2.7)
3.Evolution of density matrix elements The time evolution of the system under consideration (N=2 case) coupled to a squeezed reservoir is determined by the equations of evolution of the density matrix (eq. (2.3) ), which we have solved numerically. However, the most interesting effects of non-zero v are revealed in the limit of a strong driving field, i.e. for I a I x= v, y. An analytic description of these features is provided by the solution of the master equation in the secular approximation. For this purpose we describe the interaction picture by defining W=exp(
.
-iEJOt)pexp(iHOt)
(3.1)
It is quite simple to show that in the secular approximation the following equation of motions:
the diagonal
K2-Kv+2v2 K2
+Y2
K2-Kv-2v2 (~21wln)+(h-~2)~
K2+Kv+2v2 K2
+Y2
K'+Kv-2v2, K2 >
These equations
can be solved trivially
r1=(2fi+l)Y,
elements
of density
matrix W obey
>
Yz=(M+M*b.
(3.2)
to obtain
K2-Kv+4v2 K2
z+Yl ty
+y L
) (K+v)(K-~v) 2
K2 (3.3)
in which z is the Laplace variable The expression of n(z) reads
+
(71 +Y212
K3 and its roots
364
and we have made use of the condition
2v(K+v)(K-2v)+(K2-Kv)(3K+v)+
&$3K+v) 1 2
Tr( W) = 1 in writing down eq. (3.3
2 >
)
(3.4)
1December 1992
OPTICS COMMUNICATIONS
Volume 94, number 5
Z 1,2=-[2(y,+y2)+(2V21K2)Y11$-~~,
(3.5)
determine the decay constants of the time evolution of the diagonal elements. Note that these roots are dependent on the phase parameter of the squeezed vacuum and for a certain value of phase, these may be bigger or smaller than the thermal bath decay constants. However, the nature of these roots is given by the discriminant Y, which reads ye 1
4K4+ 12K2v2+ 16v3K K4
2
(Y1 +y#+
$gy:+
fy
(5K-4v)yi
(r,
+~2)
(3.6)
. >
It can be noticed that Y> 0 always, so the roots are real irrespective of any value of the parameters of the squeezed bath. However, the absolute value of Y would be different for various values of squeezing parameters. similarly, the off-diagonal elements of the density operators evolve in the secular approximation as $
[
wly/z)=W,2=
-
(v
-
-+I
(K-
2+-$-
[( WIy/,)=ti3,=
v)~
+12)-h]w,,
Y1 +
1
-
2+7
[(
(K+
v)’ 2
>
(K-v)’ ~
Y2
0:
Y, +
(K+ ~Yz
0:
v)~
1 1
w12
W31
K2+2(h
+Y2)
exp( 1
K2+2h
+y2)
v2
r
W3,
WI,. 2
Interestingly, evolution of W23 is uncoupled to the rest of the elements. Also, it is straightforward the dynamical evolution of WL2 and W3, is given by the following decay constants x1,2=
$1
2ivy’ iI2 .!!t 3y’2_ K _ ? + F + 4y’2_ v2_ K2 7 Y’ =Y1 ( >
For the high fields ( 16 1cx1’ B v2) we can further approximate x1,2=
3Yl _ ?
+ y.
* +
(4y’2-
v2) 112.
-Y2
.
,
v2
~exp(W
1
-ivt)
2
(3.7) to show that
(3.8)
eq. (3.8) as (3.9)
It is clear that these decay constants are complex in general. The term in the paranthesis could be real or imaginary depending on whether 2~’ > v or < v. Note that when v B y’, the decay constants would be containing a slow oscillatory part of frequency v along with other real terms. Also, X ,,2 are very sensitive to the phase of the squeezed vacuum through the parameter y2. Thus decay constants of both diagonal as well as off-diagonal elements of the density matrix W(t) are dependent on the phase parameter $J of the squeezed reservoir. For a particular choice of @the decay constants may be larger or smaller in comparison with their values in the thermal bath. This is quite obvious as y2> 0 for @= 0, and y2 < 0 for @= 7~.Hence the dynamical evolution of the Dicke system interacting with a broadband squeezed bath may be slower or faster than the thermal bath. We would pictorially depict this fact when the numerical solution of the master equation is discussed in the text. It can be shown that the expression of the expectation values of any of the dynamic variables of this system i.e. (S,) or (S+S_ ) would be containing some dc terms with coefficients WI, (t), W2, (t), W,,(t) and some rapidly oscillating terms of frequencies (A, -A,), (,l,-J,) and (1,-L,) with coefficients W,,(t), W23(t), W3, (t), respectively [ 111. Since all the coefficients WjJ are phase sensitive (as discussed above), (S,) would also depend on the phase parameter @ The evolution of W,,(t) and W,, (t) contains a slowly oscillating term at frequency v, hence the rapidly oscillating terms would be having an envelope varying at the frequency v, implying the presence of the modulation. We present these features in figs. 1, 2 and 3 which are plots of (S,) in the high field obtained by numerical solution of the master equation (2.3). To obtain exact numerical results we cast the master equation (2.3) in 365
Volume 94, number 5
OPTICS COMMUNICATIONS
t
i
Fig. 1. Time evolution of normalized (S,) in a thermal bath with ~=0,5,21aI/y=50andy/y=lO,y=l, (&),=1/2.
a c-number form by taking the matrix elements 0, - 1). The resulting equation reads as (dldt)X(t)
+M, X(t)
>
1 December 1992
Fig. 2. Same as fig. 1 but for the squeezed bath with @=O.
of the density operator p between the atomic states Ik) (k= 1,
(3.10)
where X(t) is a column vector with nine components (X1 =pl ,, X2=p10, pl_ 1, . .. etc.) and M, is a 9X 9 matrix whose matrix elements can be easily obtained after some simple operations. The general solution of the matrix equation (3.10) may be written in the form (3.11) where uk and pk are respectively the eigenvectors of M, and A, corresponding to the eigenvalue Ak. Here vk and A, are the transpose of vk and M,, respectively. The coefficients ck= ( pkk.u,) are the normalization constants. These eigenvalues and eigenvectors are easily obtained by using any standard numerical algorithm on a computer. Once X(t) (i.e. p(t) ) is known the expectation value of any atomic operator 8 can be easily expressed as (6) =Tr(@(t)). In fig. 1, we have plotted (S,) for an ordinary thermal bath ( n = 0.5 ) for v = 10 and 2 1a 1= 50. The evolution of (S,) exhibits modulation or the beating phenomenon. We have verified separately that this beating disappears when we put v= 0. The effects of squeezing are presented in figs. 2 and 3 for the phase @=O and n, respectively. The decay is clearly slower in the case of @=O and faster in the case of $= n as compared to the thermal bath, because the decay constants are phase sensitive. In other words, the modulation phenomenon is streched in one case whereas it is subdued in the other case as the decaying of the population is slower for @=O and faster for @=n in comparision to the thermal bath. There would be change in the decay constants as well as in the frequency of modulation, in general, with the change in the phase of squeezed vacuum. The physical interpretation of the above process is simple in terms of the photon field scattered amplitude in the dressed state representation. The applicability of the dressed-atom approach for a two-level atom strongly driven by a laser field, proved to be very successful in explaining the amplitudes and the frequencies of the scattered photons in the resonance fluorescence radiation of this system [ 121. For the sake of completeness here, we sketch a brief review of the dressed state picture. In the dressed state picture, one has to consider the system formed by the atom and the laser photons. For a two-level atom, the unperturbed states le, n) and Ig, n+ 1) (i.e. the atom in the excited state e or ground state g in the presence of n or n+ 1 photons) are 366
OPTICS COMMUNICATIONS
Volume 94, number 5
Y
k? d ‘0.00
0.50
1.00
1.50
t
2.00
2.50
Fig. 3. Same as fig. 1 but for the squeezed bath with o= n.
1 December 1992
Fig. 4. Schematic dressed states diagram showing the n and n - 1 multiplets of the dressed two-level atom (a) and the dressed two (two-level) atom system (b) together with some of the possible downward spontaneous transitions. For (a), the transitions A, B, C contribute to the photon scattered at frequencies wO,00+2a, w,,- 2cu,respectively and for (b ) the transitions A, B and C contribute to the photon emitted at o,,, w0+(u+K)/2, or,(v+K)/2, respectively.
separated by the detuning A= coo-o =. Now the interaction between the atom and the photons introduces a coupling that leads to the dressed-atom energy levels (fig. 4a). The two states I 1, n) and 12, n), which are some linear combination of 1e, n ) and 1g, n = 1) are separated by a splitting CY(if A= 0)) the Rabi frequency. For an intense field the resonance fluorescence can be described as being due to the spontaneous transition between the dressed atom levels. Some of the allowed transitions which correspond to the non-zero matrix elements of the atomic dipole moment are depicted in fig. 4a. There are definite correlations between the emitted photons at these allowed transitions [ 121. Thus if we plot intensity Z(t) of the fluorescence versus time, we observe oscillations of the intensity at the frequency 2o as a result of interference between these scattered amplitudes. Hence it explains the Rabi oscillations phenomenon for a strongly driven two-level atom. Next, we extend this interpretation to a system of two two-level atoms. Figure 4b represents the dressed energy states diagram for two adjacent multiplets. In the case of the two atom system the three states I 1, n) , 12, n) and 13, n) are some linear combination of the triplet components I - 1, n), 10, n) and ) 1, n) and are unequally separated due to non-zero dipole-dipole interaction between atoms. However, for non-interacting atoms ( v = 0)) the separation is equal and given by 2o. It is apparent that the possibility of fluorescence exists at seven different frequencies wo, o. ? (Y-K) /2, w. + (Y + K) /2. All these scattered amplitudes at various frequencies have some definite correlation between them and as a result of interference between frequencies o. + (V-K) /2, o. - ( v + K) /2, etc., one gets a beating phenomenon of frequency v in the usual Rabi oscillations in the fluorescence intensity. It is straightforward to see that when v=O the two Rabi frequencies just coincide and thus beating disappears. It is quite clear from the figures that the atomic system relaxes into a state of equilibrium with a modified phase dependent decay coefficient showing typical super-radiant behaviour of the Dicke model. Though the beating phenomenon is present in the thermal bath, the presence of a squeezed bath makes it phase sensitive. AS a consequence of the modulation, there is a situation similar to temporal hole burning (i.e. redistribution) in the population of the decay system. Thus we have presented the behaviour of population inversion in the two atom Dicke model with the dipoledipole interaction included, and observed a modulation phenomenon. We have also obtained the analytic expression of various decay constants involved in the population decay and explained the modulation phenomenon by isolating the slow oscillatory part in the decay constants. The results presented here may be useful experimentally to observe the presence of the dipole-dipole interaction between the two atoms and also in 367
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5
OPTICS
I December
COMMUNICATIONS
1992
determining the strength of the dipole interaction. There is the possibility of realization of the dynamics discussed here in the experiments with two trapped ions [ 131 or with two atoms fixed either in organic layers [ 141 etc., which may enable one to get better understanding of the multi-atom effects.
References [ 1 ] C.W. Gardiner, Phys. Rev. Lett. 56 (1986) 1917. [2] H.J. Carmichael, AS. Lane and D.F. Walls, Phys. Rev. Lett. 58 (1987) 2539; J. Mod. Optics 34 (1987) [3] G.S. Agatwal and R.R. Puri, Optics Comm. 69 (1990) 269. [4] G.S. Agarwal and R.R. Puri, Phys. Rev. A 41 (1990) 3782. [ 51 G. Palma and P.L. Knight, Phys. Rev. A 39 ( 1989) 1462. [6] G.S. Agarwal, Springer Tracts in Modem Physics vol. 70, ed. G. Hohler (Springer, Berlin, 1974). [ 71 R.K. Bullough et al., paper presented at ICQE, Anaheim, 1989. [ 81 R.R. Puri, A. Joshi and R.K. Bullough, Int. J. Mod. Phys. B 5 ( 199 1) 3 I 15. [ 91 M. Gross and S. Haroche, Phys. Rep. 93 ( 1982) 301. [lo] G.S. Agarwal, L.M. Narducci and E. Apostolidis, Optics Comm. 36 (1981) 285. [ 111 R.R. Puri and G.S. Agatwal, Phys. Rev. A 35 (1987) 3433. [ 121 C. Cohen-Tannoudji and S. Reynaud, Philos. Trans. R. Sot. London Ser. A 239 ( 1979) 223; R. Loudon, in: Quantum theory of light (Oxford Univ. Press, Oxford, 1973). [ 131 W. Neuhauser and T. Sauter, Comm. At. Mol. Phys. 21 (1988) 83. [ 141 K.H. Drexhage, Prog. Optics 12 (1974) 165.
368
821.
94 ( 1992) 362-368
OPTICS COMMUNICATIONS
Effect of dispersion forces on the dynamical evolution of the two-atom Dicke model in a broadband squeezed bath Amitabh
Joshi and R.R. Puri
MDRS and Theoretical Physics Division, Bhabha Atomic Research Centre, Trombay, Bombay 400 085. India Received
13 February
1992; revised manuscript
received
7 July 1992
We show that the two-atom Dicke model, including dipole-dipole interaction (dispersion forces) and interaction with a broadband squeezed bath, exhibits a periodic modulation phenomenon in its population decay. For this purpose we have calculated explicitly the analytic expression for the time evolution of the density matrix elements under the intense field limit. The phase sensitive nature of the modulation phenomenon has been clearly brought out in these calculations.
1. Introduction The interaction of a system of atoms with a broadband squeezed bath of radiation having modes correlated in pairs gives rise to some unusual dynamic and steady state characteristics [ l-5 1. The most striking example is the case of a strongly driven two-level atom interacting with such a squeezed bath resulting in the narrowing of the central peak of the Mollow triplet or an equality of heights of all the three peaks [ 21. The steady state of a system of two-level atoms collectively interacting with such a squeezed bath can be a pure state under certain conditions [ 3-5 ] whereas the mixed state is attained under the influence of an ordinary bath. Thus these unusual properties are a manifestation of the correlations present in the squeezed bath. Here, we explore further the effects of a squeezed bath on the dynamical properties of a system of two-level atoms. The Dicke model of two two-level atoms interacting collectively with the squeezed bath and driven by an external field is considered here. Our aim is to explore dynamic behaviour of such a system in the high field limit. In our analysis the dipole-dipole interaction (dispersion forces) term in the master equation is included. This term is important in the Dicke model which assumes close spacing between the atoms ( -A). This term originates by retaining the atomic position dependent terms in the hamiltonian and going to the limit of small systems after eliminating bath operators from the equation for the density matrix [ 61. On the contrary, if, small system limit is assumed in writing the atom-field hamiltonian itself, such a term does not arise. In sec. 2, we introduce the model and master equation for the Dicke system of N identical two-level atoms interacting with a broadband squeezed bath. We then calculate the eigenvalues and eigenstates of the two-atom system hamiltonian. In sec. 3, we derive analytic expression for the evolution of the density matrix elements in the high field limit which are useful to obtain expectation values of the dynamic variables. This is followed by discussions and concluding remarks.
2. The model We consider a system of N identical two-level atoms each of transition frequency w. interacting collectively with a broadband squeezed bath which is pumped by a coherent field of frequency wL. The atom system is also driven by an external field, which is also of frequency wi_. Further, we assume the bandwidth of the squeezed 362
0030-4018/92/$
05.00 0 1992 Elsevier Science Publishers
B.V. All rights reserved.
1 December 1992
OPTICS COMMUNICATIONS
Volume 94, number 5
bath to be broader than the spectral width of the fluorescence so that the spectrum falls well within the bandwidths of the squeezed vacuum modes. The system can then be described by the following master equation [ 731 for the reduced density matrix (in Born-Markov approximation and in a frame rotating with wL): $/at=-id[S,,p]-i[oS++o!*S_,p]-i +n(2S+pS_
1 Q,[S.?S;,p]+y[(ti+1)(2S_pS+-S+S_p-pS+S_) i#J
-s_S+p--pS_S+)+M(2S+pS+
-s:p-ps:)+w(2s_ps_
-sz_p-pS2)]
)
(2.1)
where, A= coo- co=, M= 1MI exp (iv) and ST, S, are collective atomic operators and (Y= I a I exp ( igL) is the Rabi frequency associated with the field. Here, Q, (Q,=Q,,) is the coefficient of dipole-dipole interaction. Also, IMI 2< A( A+ 1)) the equality sign holds for an ideal squeezed bath. We consider here the ideal squeezed bath conditions. The master equation is S2 conserving, provided the dipole-dipole interaction term Qij= 0. The dipole-dipole interaction term does not, in general, conserve S* except for a special symmetric arrangement of the atoms [ 9 ] or for N= 2 in which case
(2.2) where VEQ,~=Q~,.
The master equation
then reads
ap~at=-i[~,,p]+~[(~+i)(2S_pS+-S+S_p-pS+S_)+n(2S+pS_-S_S+p-pS_S+) +M(2S+pS+
-s:p-ps:)+M*(2s_ps_
-SLp-pS2_)]
)
(2.3)
and &=A&+
Ial (S, +S_)+vSS.
(2.4)
Here we have redefined Sk exp ( f igL) as Sk and M= IMI exp (i@) with $= q- 2eL. Note that the hamiltonian Ho is known as the two atom Dicke model hamiltonian with dipole-dipole interaction taken into account [ lo]. The only states taking part in the dynamics are the triplet states ( ]e) , 1g) describe the excited and ground states of the two identical atoms, respectively) (2.5a) and the singlet state (2Sb) is completely decoupled optically from the triplet state. This would be understood from eq. (2.4) which is invariant under the exchange a c* b which implies that the symmetry of the atomic state is a constant of motion. The singlet state is antisymmetric while the triplet states are symmetric, which means that they cannot be coupled by the interaction with the field. In this case it is quite simple to calculate the new frequencies of the system by diagonalization of the effective hamiltonian Ho. If A # 0, the secular equation leading to the dressed eigenfrequencies is a cubic equation that can be best solved by numerical techniques. If A=O, instead, the secular equation is still a cubic equation, which can be best solved analytically and the eigenfrequencies of the hamiltonian (2.4) are given explicitly by A,=v,
3L2,3=v/2?K/2
and the corresponding
andK=(v*+16]~1*)“*, eigenstates
(2.6)
are 363
Iv2)=D;1[2Jzlal
lr~)=~(l-)-I1w>
)W+~$[2&t
1 December 1992
OPTICS COMMUNICATIONS
Volume 94, number 5
I-1)-(K-v)10)+2fiIaI
II>1 1
I-1>+(K-w)+2JZlal ll)],
D,,z=[161a!12+(K~~)211’2.
(2.7)
3.Evolution of density matrix elements The time evolution of the system under consideration (N=2 case) coupled to a squeezed reservoir is determined by the equations of evolution of the density matrix (eq. (2.3) ), which we have solved numerically. However, the most interesting effects of non-zero v are revealed in the limit of a strong driving field, i.e. for I a I x= v, y. An analytic description of these features is provided by the solution of the master equation in the secular approximation. For this purpose we describe the interaction picture by defining W=exp(
.
-iEJOt)pexp(iHOt)
(3.1)
It is quite simple to show that in the secular approximation the following equation of motions:
the diagonal
K2-Kv+2v2 K2
+Y2
K2-Kv-2v2 (~21wln)+(h-~2)~
K2+Kv+2v2 K2
+Y2
K'+Kv-2v2
These equations
can be solved trivially
r1=(2fi+l)Y,
elements
of density
matrix W obey
>
Yz=(M+M*b.
(3.2)
to obtain
K2-Kv+4v2 K2
z+Yl ty
+y L
) (K+v)(K-~v) 2
K2 (3.3)
in which z is the Laplace variable The expression of n(z) reads
+
(71 +Y212
K3 and its roots
364
and we have made use of the condition
2v(K+v)(K-2v)+(K2-Kv)(3K+v)+
&$3K+v) 1 2
Tr( W) = 1 in writing down eq. (3.3
2 >
)
(3.4)
1December 1992
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Volume 94, number 5
Z 1,2=-[2(y,+y2)+(2V21K2)Y11$-~~,
(3.5)
determine the decay constants of the time evolution of the diagonal elements. Note that these roots are dependent on the phase parameter of the squeezed vacuum and for a certain value of phase, these may be bigger or smaller than the thermal bath decay constants. However, the nature of these roots is given by the discriminant Y, which reads ye 1
4K4+ 12K2v2+ 16v3K K4
2
(Y1 +y#+
$gy:+
fy
(5K-4v)yi
(r,
+~2)
(3.6)
. >
It can be noticed that Y> 0 always, so the roots are real irrespective of any value of the parameters of the squeezed bath. However, the absolute value of Y would be different for various values of squeezing parameters. similarly, the off-diagonal elements of the density operators evolve in the secular approximation as $
[
wly/z)=W,2=
-
(v
-
-+I
(K-
2+-$-
[( WIy/,)=ti3,=
v)~
+12)-h]w,,
Y1 +
1
-
2+7
[(
(K+
v)’ 2
>
(K-v)’ ~
Y2
0:
Y, +
(K+ ~Yz
0:
v)~
1 1
w12
W31
K2+2(h
+Y2)
exp( 1
K2+2h
+y2)
v2
r
W3,
WI,. 2
Interestingly, evolution of W23 is uncoupled to the rest of the elements. Also, it is straightforward the dynamical evolution of WL2 and W3, is given by the following decay constants x1,2=
$1
2ivy’ iI2 .!!t 3y’2_ K _ ? + F + 4y’2_ v2_ K2 7 Y’ =Y1 ( >
For the high fields ( 16 1cx1’ B v2) we can further approximate x1,2=
3Yl _ ?
+ y.
* +
(4y’2-
v2) 112.
-Y2
.
,
v2
~exp(W
1
-ivt)
2
(3.7) to show that
(3.8)
eq. (3.8) as (3.9)
It is clear that these decay constants are complex in general. The term in the paranthesis could be real or imaginary depending on whether 2~’ > v or < v. Note that when v B y’, the decay constants would be containing a slow oscillatory part of frequency v along with other real terms. Also, X ,,2 are very sensitive to the phase of the squeezed vacuum through the parameter y2. Thus decay constants of both diagonal as well as off-diagonal elements of the density matrix W(t) are dependent on the phase parameter $J of the squeezed reservoir. For a particular choice of @the decay constants may be larger or smaller in comparison with their values in the thermal bath. This is quite obvious as y2> 0 for @= 0, and y2 < 0 for @= 7~.Hence the dynamical evolution of the Dicke system interacting with a broadband squeezed bath may be slower or faster than the thermal bath. We would pictorially depict this fact when the numerical solution of the master equation is discussed in the text. It can be shown that the expression of the expectation values of any of the dynamic variables of this system i.e. (S,) or (S+S_ ) would be containing some dc terms with coefficients WI, (t), W2, (t), W,,(t) and some rapidly oscillating terms of frequencies (A, -A,), (,l,-J,) and (1,-L,) with coefficients W,,(t), W23(t), W3, (t), respectively [ 111. Since all the coefficients WjJ are phase sensitive (as discussed above), (S,) would also depend on the phase parameter @ The evolution of W,,(t) and W,, (t) contains a slowly oscillating term at frequency v, hence the rapidly oscillating terms would be having an envelope varying at the frequency v, implying the presence of the modulation. We present these features in figs. 1, 2 and 3 which are plots of (S,) in the high field obtained by numerical solution of the master equation (2.3). To obtain exact numerical results we cast the master equation (2.3) in 365
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t
i
Fig. 1. Time evolution of normalized (S,) in a thermal bath with ~=0,5,21aI/y=50andy/y=lO,y=l, (&),=1/2.
a c-number form by taking the matrix elements 0, - 1). The resulting equation reads as (dldt)X(t)
+M, X(t)
>
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Fig. 2. Same as fig. 1 but for the squeezed bath with @=O.
of the density operator p between the atomic states Ik) (k= 1,
(3.10)
where X(t) is a column vector with nine components (X1 =pl ,, X2=p10, pl_ 1, . .. etc.) and M, is a 9X 9 matrix whose matrix elements can be easily obtained after some simple operations. The general solution of the matrix equation (3.10) may be written in the form (3.11) where uk and pk are respectively the eigenvectors of M, and A, corresponding to the eigenvalue Ak. Here vk and A, are the transpose of vk and M,, respectively. The coefficients ck= ( pkk.u,) are the normalization constants. These eigenvalues and eigenvectors are easily obtained by using any standard numerical algorithm on a computer. Once X(t) (i.e. p(t) ) is known the expectation value of any atomic operator 8 can be easily expressed as (6) =Tr(@(t)). In fig. 1, we have plotted (S,) for an ordinary thermal bath ( n = 0.5 ) for v = 10 and 2 1a 1= 50. The evolution of (S,) exhibits modulation or the beating phenomenon. We have verified separately that this beating disappears when we put v= 0. The effects of squeezing are presented in figs. 2 and 3 for the phase @=O and n, respectively. The decay is clearly slower in the case of @=O and faster in the case of $= n as compared to the thermal bath, because the decay constants are phase sensitive. In other words, the modulation phenomenon is streched in one case whereas it is subdued in the other case as the decaying of the population is slower for @=O and faster for @=n in comparision to the thermal bath. There would be change in the decay constants as well as in the frequency of modulation, in general, with the change in the phase of squeezed vacuum. The physical interpretation of the above process is simple in terms of the photon field scattered amplitude in the dressed state representation. The applicability of the dressed-atom approach for a two-level atom strongly driven by a laser field, proved to be very successful in explaining the amplitudes and the frequencies of the scattered photons in the resonance fluorescence radiation of this system [ 121. For the sake of completeness here, we sketch a brief review of the dressed state picture. In the dressed state picture, one has to consider the system formed by the atom and the laser photons. For a two-level atom, the unperturbed states le, n) and Ig, n+ 1) (i.e. the atom in the excited state e or ground state g in the presence of n or n+ 1 photons) are 366
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Y
k? d ‘0.00
0.50
1.00
1.50
t
2.00
2.50
Fig. 3. Same as fig. 1 but for the squeezed bath with o= n.
1 December 1992
Fig. 4. Schematic dressed states diagram showing the n and n - 1 multiplets of the dressed two-level atom (a) and the dressed two (two-level) atom system (b) together with some of the possible downward spontaneous transitions. For (a), the transitions A, B, C contribute to the photon scattered at frequencies wO,00+2a, w,,- 2cu,respectively and for (b ) the transitions A, B and C contribute to the photon emitted at o,,, w0+(u+K)/2, or,(v+K)/2, respectively.
separated by the detuning A= coo-o =. Now the interaction between the atom and the photons introduces a coupling that leads to the dressed-atom energy levels (fig. 4a). The two states I 1, n) and 12, n), which are some linear combination of 1e, n ) and 1g, n = 1) are separated by a splitting CY(if A= 0)) the Rabi frequency. For an intense field the resonance fluorescence can be described as being due to the spontaneous transition between the dressed atom levels. Some of the allowed transitions which correspond to the non-zero matrix elements of the atomic dipole moment are depicted in fig. 4a. There are definite correlations between the emitted photons at these allowed transitions [ 121. Thus if we plot intensity Z(t) of the fluorescence versus time, we observe oscillations of the intensity at the frequency 2o as a result of interference between these scattered amplitudes. Hence it explains the Rabi oscillations phenomenon for a strongly driven two-level atom. Next, we extend this interpretation to a system of two two-level atoms. Figure 4b represents the dressed energy states diagram for two adjacent multiplets. In the case of the two atom system the three states I 1, n) , 12, n) and 13, n) are some linear combination of the triplet components I - 1, n), 10, n) and ) 1, n) and are unequally separated due to non-zero dipole-dipole interaction between atoms. However, for non-interacting atoms ( v = 0)) the separation is equal and given by 2o. It is apparent that the possibility of fluorescence exists at seven different frequencies wo, o. ? (Y-K) /2, w. + (Y + K) /2. All these scattered amplitudes at various frequencies have some definite correlation between them and as a result of interference between frequencies o. + (V-K) /2, o. - ( v + K) /2, etc., one gets a beating phenomenon of frequency v in the usual Rabi oscillations in the fluorescence intensity. It is straightforward to see that when v=O the two Rabi frequencies just coincide and thus beating disappears. It is quite clear from the figures that the atomic system relaxes into a state of equilibrium with a modified phase dependent decay coefficient showing typical super-radiant behaviour of the Dicke model. Though the beating phenomenon is present in the thermal bath, the presence of a squeezed bath makes it phase sensitive. AS a consequence of the modulation, there is a situation similar to temporal hole burning (i.e. redistribution) in the population of the decay system. Thus we have presented the behaviour of population inversion in the two atom Dicke model with the dipoledipole interaction included, and observed a modulation phenomenon. We have also obtained the analytic expression of various decay constants involved in the population decay and explained the modulation phenomenon by isolating the slow oscillatory part in the decay constants. The results presented here may be useful experimentally to observe the presence of the dipole-dipole interaction between the two atoms and also in 367
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determining the strength of the dipole interaction. There is the possibility of realization of the dynamics discussed here in the experiments with two trapped ions [ 131 or with two atoms fixed either in organic layers [ 141 etc., which may enable one to get better understanding of the multi-atom effects.
References [ 1 ] C.W. Gardiner, Phys. Rev. Lett. 56 (1986) 1917. [2] H.J. Carmichael, AS. Lane and D.F. Walls, Phys. Rev. Lett. 58 (1987) 2539; J. Mod. Optics 34 (1987) [3] G.S. Agatwal and R.R. Puri, Optics Comm. 69 (1990) 269. [4] G.S. Agarwal and R.R. Puri, Phys. Rev. A 41 (1990) 3782. [ 51 G. Palma and P.L. Knight, Phys. Rev. A 39 ( 1989) 1462. [6] G.S. Agarwal, Springer Tracts in Modem Physics vol. 70, ed. G. Hohler (Springer, Berlin, 1974). [ 71 R.K. Bullough et al., paper presented at ICQE, Anaheim, 1989. [ 81 R.R. Puri, A. Joshi and R.K. Bullough, Int. J. Mod. Phys. B 5 ( 199 1) 3 I 15. [ 91 M. Gross and S. Haroche, Phys. Rep. 93 ( 1982) 301. [lo] G.S. Agarwal, L.M. Narducci and E. Apostolidis, Optics Comm. 36 (1981) 285. [ 111 R.R. Puri and G.S. Agatwal, Phys. Rev. A 35 (1987) 3433. [ 121 C. Cohen-Tannoudji and S. Reynaud, Philos. Trans. R. Sot. London Ser. A 239 ( 1979) 223; R. Loudon, in: Quantum theory of light (Oxford Univ. Press, Oxford, 1973). [ 131 W. Neuhauser and T. Sauter, Comm. At. Mol. Phys. 21 (1988) 83. [ 141 K.H. Drexhage, Prog. Optics 12 (1974) 165.
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